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Algebraic number field

An algebraic number field is a finite field extension K of the rational numbers \mathbb{Q}, with degree n = [K : \mathbb{Q}] \geq 1, where n = 1 corresponds to \mathbb{Q} itself.^[1] Such fields are typically generated by adjoining a single algebraic number \alpha, so K = \mathbb{Q}(\alpha), where \alpha is a root of an irreducible monic polynomial of degree n with rational coefficients.^[2] Equivalently, K can be formed by the roots of an irreducible polynomial with integer coefficients, making it a key object in algebraic number theory for studying arithmetic properties beyond the rationals.^[3] Every algebraic number field K admits n distinct embeddings into the complex numbers \mathbb{C}, consisting of r real embeddings and s pairs of complex conjugate embeddings, satisfying n = r + 2s.^[1] The ring of integers \mathcal{O}_K of K, defined as the integral closure of \mathbb{Z} in K, comprises all algebraic integers in K and forms a Dedekind domain: it is Noetherian, integrally closed, and every nonzero prime ideal is maximal, enabling unique factorization of ideals into prime ideals despite the general failure of unique element factorization.^[1] The discriminant \Delta_K of K is a fundamental integer invariant that encodes ramification information for prime ideals and satisfies properties such as (-1)^s for its sign and congruence to 0 or 1 modulo 4.^[1] Algebraic number fields underpin much of modern number theory, including the study of units in \mathcal{O}_K, which form a finitely generated abelian group of rank r + s - 1 with torsion subgroup given by the roots of unity in K; the ideal class group, whose order (the class number h_K) is finite and bounded by the Minkowski constant; and extensions like Hilbert class fields that resolve ideal class issues.^[1] They also feature a product formula for norms, stating that for any nonzero \alpha \in K, the product of its absolute values over all places equals 1, generalizing the rationals' properties.^[1]

Definition and Basics

Prerequisites

In field theory, a field extension K/\mathbb{Q} is a pair consisting of a field K and an embedding of the rational numbers \mathbb{Q} into K as a subfield, where K is viewed as a vector space over \mathbb{Q}.^[4] An extension K/\mathbb{Q} is algebraic if every element \alpha \in K is algebraic over \mathbb{Q}, meaning there exists a non-constant polynomial f(x) \in \mathbb{Q} such that f(\alpha) = 0.^[4] Such extensions are central to algebraic number theory, as they provide the framework for studying numbers beyond the rationals that satisfy polynomial equations with rational coefficients.^[4] A finite extension K/\mathbb{Q} is one in which the dimension of K as a vector space over \mathbb{Q} is finite, denoted by the degree [K:\mathbb{Q}] = n < \infty.^[4] In this context, key terminology includes algebraic numbers, which are the elements of algebraic extensions of \mathbb{Q} embedded in the complex numbers \mathbb{C}.^[4] For an algebraic number \alpha, the minimal polynomial \mathrm{irr}(\alpha, \mathbb{Q}) is the unique monic polynomial in \mathbb{Q} of least degree such that \mathrm{irr}(\alpha, \mathbb{Q})(\alpha) = 0, and the degree of this polynomial equals n = [\mathbb{Q}(\alpha):\mathbb{Q}].^[4] Algebraic extensions K/\mathbb{Q} are further classified by separability: an extension is separable if the minimal polynomial of every \alpha \in K over \mathbb{Q} has distinct roots in a splitting field (or algebraic closure) of \mathbb{Q}.^[5] Since \mathbb{Q} has characteristic zero, every algebraic extension of \mathbb{Q} is automatically separable.^[5] This property ensures that the structure of such extensions is well-behaved, avoiding complications from multiple roots that arise in positive characteristic.^[5] The foundational development of algebraic number fields traces back to Richard Dedekind's work in the late 19th century, where he introduced the concept of algebraic integers and ideals to resolve issues in unique factorization within these extensions, as detailed in his 1871 supplement to Dirichlet's Vorlesungen über Zahlentheorie.^[6]

Definition

An algebraic number field, or simply a number field, is a finite-degree field extension of the rational numbers \mathbb{Q}. Specifically, if K is a number field, then there exists a finite set of algebraic numbers \alpha_1, \dots, \alpha_r \in \overline{\mathbb{Q}} such that K = \mathbb{Q}(\alpha_1, \dots, \alpha_r), where \overline{\mathbb{Q}} denotes the algebraic closure of \mathbb{Q}, and the degree of the extension is n = [K : \mathbb{Q}] < \infty.^[7] As a field extension of finite degree, K is a vector space over \mathbb{Q} of dimension n.^[7] Since \mathbb{Q} has characteristic zero, every finite extension K/\mathbb{Q} is separable, and thus by the primitive element theorem, K admits a simple presentation: there exists a primitive element \alpha \in K such that K = \mathbb{Q}(\alpha), where \alpha is algebraic over \mathbb{Q} of degree n.^[8] The minimal polynomial of \alpha over \mathbb{Q} is then an irreducible monic polynomial f(x) \in \mathbb{Q} of degree n, and K \cong \mathbb{Q}/(f(x)).^[8] A number field K of degree n over \mathbb{Q} admits exactly n distinct embeddings \sigma_i : K \hookrightarrow \mathbb{C}, i=1,\dots,n, into the complex numbers; these embeddings are determined by sending the primitive element \alpha to one of its n roots in \mathbb{C}.^[9] Each such embedding is either real (mapping into \mathbb{R}) or complex (with a complex conjugate pair). The elements of K are algebraic numbers over \mathbb{Q}.^[7]

Examples and Properties

Standard Examples

One of the most basic examples of an algebraic number field is the quadratic field \mathbb{Q}(\sqrt{d}), where d is a square-free integer not equal to $0

or &#36;1

. This field is generated over \mathbb{Q} by adjoining \sqrt{d}, which satisfies the minimal polynomial x^2 - d = 0, and thus has degree $2over\mathbb{Q}.[15] Real quadratic fields arise when d > 0, such as \mathbb{Q}(\sqrt{2})or\mathbb{Q}(\sqrt{5}), while imaginary quadratic fields occur for d < 0, like \mathbb{Q}(\sqrt{-1})or\mathbb{Q}(\sqrt{-3})$.^[10] Another standard class consists of cyclotomic fields \mathbb{Q}(\zeta_m), where \zeta_m = e^{2\pi i / m} is a primitive m-th root of unity and m \geq 3 is an integer. These fields are Galois extensions of \mathbb{Q} of degree \phi(m), where \phi denotes Euler's totient function; for instance, \mathbb{Q}(\zeta_3) has degree $2, \mathbb{Q}(\zeta_4) = \mathbb{Q}(i)

has degree &#36;2

, and \mathbb{Q}(\zeta_5) has degree $4.[68] Cyclotomic fields play a central role in class field theory as they generate all abelian extensions of \mathbb{Q}$.^[11] For higher degrees, pure cubic fields provide simple illustrations, such as

\mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})

, which is generated by a real cube root of $2satisfying the minimal polynomialx^3 - 2 = 0

and thus has degree &#36;3

over \mathbb{Q}.^[12] This field is not Galois over \mathbb{Q}, as its splitting field requires adjoining a primitive cube root of unity as well.^[13] In quadratic fields \mathbb{Q}(\sqrt{d}), the discriminant \Delta is computed as \Delta = 4d when d \equiv 2,3 \pmod{4}, and \Delta = d when d \equiv 1 \pmod{4}; for example, \Delta = 8 for \mathbb{Q}(\sqrt{2}) and \Delta = 5 for \mathbb{Q}(\sqrt{5}).^[14] This invariant measures the "ramification" at finite primes and distinguishes non-isomorphic fields. For d \equiv 2,3 \pmod{4}, the ring of integers is \mathbb{Z}[\sqrt{d}].^[15] Algebraic number fields are often classified by their signature (r_1, r_2), where r_1 is the number of real embeddings into \mathbb{R} and r_2 is the number of pairs of complex conjugate embeddings, satisfying n = r_1 + 2r_2 with n = [\mathbb{Q}(\alpha):\mathbb{Q}].^[16] Totally real fields have r_1 = n and r_2 = 0, such as real quadratic fields; totally complex fields have r_1 = 0, like cyclotomic fields for m \geq 3; and CM (complex multiplication) fields are totally imaginary quadratic extensions of totally real fields, including all imaginary quadratic fields.^[17]

Non-Examples

While algebraic number fields are characterized by being finite-degree extensions of the rational numbers \mathbb{Q}, certain field extensions fail to qualify due to violations of this finiteness condition or the algebraicity requirement.^[18] Infinite algebraic extensions, such as the algebraic closure \overline{\mathbb{Q}} of \mathbb{Q}, consist entirely of algebraic numbers but possess infinite degree over \mathbb{Q}, as they adjoin roots of all irreducible polynomials over \mathbb{Q} simultaneously, resulting in an uncountably infinite-dimensional vector space.^[18] This infinitude precludes \overline{\mathbb{Q}} from being an algebraic number field, distinguishing it from finite cases like quadratic extensions.^[18] Transcendental extensions, exemplified by \mathbb{Q}(\pi) or the real numbers \mathbb{R}, incorporate elements that are not roots of any non-zero polynomial with rational coefficients, thereby lacking the full algebraicity over \mathbb{Q} essential to number fields.^[19] In \mathbb{Q}(\pi), \pi itself is transcendental, ensuring the extension has transcendence degree 1 and is not algebraic.^[19] Similarly, \mathbb{R} contains uncountably many transcendental elements, rendering it an improper infinite transcendental extension over \mathbb{Q}.^[19] Function fields, such as the rational function field \mathbb{Q}(x), arise as fields of rational functions in one indeterminate over \mathbb{Q} and exhibit transcendental behavior with infinite transcendence degree, contrasting sharply with the algebraic, finite-dimensional structure of number fields.^[20] These fields model arithmetic on algebraic curves and possess non-trivial derivations absent in number fields, underscoring their distinct geometric origins.^[20] p-adic fields like \mathbb{Q}_p, the completion of \mathbb{Q} with respect to the p-adic valuation for a prime p, are infinite extensions that are not algebraic over \mathbb{Q}, as they include limit points not satisfying polynomial equations with rational coefficients.^[18] Instead, \mathbb{Q}_p serves as a local field, providing a completion rather than a finite algebraic extension.^[18]

Ring of Integers

Definition and Construction

The ring of integers \mathcal{O}_K of an algebraic number field K, which is a finite extension of \mathbb{Q}, is defined as the set of all algebraic integers in K. Specifically, \mathcal{O}_K = \{\alpha \in K \mid \alpha is an algebraic integer\}, where an algebraic integer \alpha is an element whose minimal polynomial over \mathbb{Q} is monic and has coefficients in \mathbb{Z}.^[18] This set forms a subring of K that serves as the integral closure of \mathbb{Z} in K, meaning every element of \mathcal{O}_K satisfies a monic polynomial equation with coefficients in \mathbb{Z}, and \mathcal{O}_K contains all such elements from K.^[21] As the integral closure, \mathcal{O}_K is integrally closed in K, ensuring no larger subring of K containing \mathbb{Z} consists entirely of algebraic integers.^[18] Moreover, \mathcal{O}_K is a Dedekind domain, characterized by being an integrally closed Noetherian domain of Krull dimension one, which underpins its role in the arithmetic of number fields.^[22] A fundamental theorem states that for any number field K, the ring of integers \mathcal{O}_K is unique up to isomorphism as the maximal order in K.^[18] This uniqueness follows from the fact that the integral closure of \mathbb{Z} in K is the largest subring where every element is integral over \mathbb{Z}.^[23] Explicit constructions are available for certain fields, such as quadratic number fields K = \mathbb{Q}(\sqrt{d}) where d is a square-free integer. If d \equiv 2 \pmod{4} or d \equiv 3 \pmod{4}, then \mathcal{O}_K = \mathbb{Z}[\sqrt{d}]; otherwise, if d \equiv 1 \pmod{4}, then \mathcal{O}_K = \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right].^[15] These forms arise by verifying which elements satisfy monic polynomials with integer coefficients, confirming they exhaust the algebraic integers in K.^[24] The concept of the ring of integers was motivated by Ernst Kummer's development of ideal numbers in the mid-19th century, introduced to address the failure of unique factorization in the rings of integers of cyclotomic fields while attempting proofs related to Fermat's Last Theorem.^[25] Kummer's work highlighted the need for a systematic structure like \mathcal{O}_K to restore unique factorization via ideals.^[26]

Unique Factorization and Ideals

The ring of integers \mathcal{O}_K of an algebraic number field K is a Dedekind domain, meaning it is a Noetherian integral domain that is integrally closed in its fraction field and in which every nonzero prime ideal is maximal.^[27] As a Noetherian ring, every ideal in \mathcal{O}_K is finitely generated as a \mathbb{Z}-module.^[27] Integrally closed means that \mathcal{O}_K contains all elements of K that are integral over \mathbb{Z}, ensuring no larger ring of integers exists within K.^[27] Ideals in \mathcal{O}_K are nonzero \mathbb{Z}-submodules of \mathcal{O}_K that are closed under multiplication by elements of \mathcal{O}_K.^[28] A principal ideal is one generated by a single element, such as (a) = \{ a \cdot \beta \mid \beta \in \mathcal{O}_K \} for a \in \mathcal{O}_K.^[27] However, not all ideals are principal; non-principal ideals arise when unique factorization fails for elements of \mathcal{O}_K, though the ring may still possess unique factorization in terms of ideals.^[28] A fundamental theorem states that every nonzero ideal in \mathcal{O}_K factors uniquely as a product of prime ideals: for any nonzero ideal \mathfrak{a}, there exist distinct prime ideals \mathfrak{p}_i and positive integers e_i such that \mathfrak{a} = \prod \mathfrak{p}_i^{e_i}, with uniqueness up to ordering of the factors.^[27] In particular, for a principal ideal (a), the factorization is (a) = \prod \mathfrak{p}_i^{e_i}.^[28] This ideal-theoretic unique factorization holds even when \mathcal{O}_K is not a unique factorization domain for its elements.^[27] The extent to which unique element factorization fails is measured by the ideal class group \mathrm{Cl}_K, defined as the group of fractional ideals of \mathcal{O}_K modulo the subgroup of principal fractional ideals, under multiplication.^[27] The order of \mathrm{Cl}_K, known as the class number h_K, is finite and equals 1 if and only if every ideal is principal (i.e., \mathcal{O}_K is a principal ideal domain).^[27] Nontrivial classes in \mathrm{Cl}_K correspond to non-principal ideals that are not equivalent to principal ones via multiplication by units.^[29] A concrete example occurs in the field K = \mathbb{Q}(\sqrt{-5}), where \mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]. The principal ideal (2) factors as \mathfrak{p}^2, where \mathfrak{p} = (2, 1 + \sqrt{-5}) is a non-principal prime ideal of norm 2.^[15] This factorization is unique, but since \mathfrak{p} is non-principal, the class number h_K = 2, indicating that unique element factorization fails (e.g., $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}) represents two distinct factorizations up to units).^[15]

Bases and Modules

Integral Basis

In an algebraic number field K of degree n = [K : \mathbb{Q}], an integral basis for the ring of integers \mathcal{O}_K is a set \{\omega_1, \dots, \omega_n\} \subset \mathcal{O}_K such that every element of \mathcal{O}_K can be uniquely expressed as an integer linear combination \sum_{i=1}^n a_i \omega_i with a_i \in \mathbb{Z}, and \{\omega_1, \dots, \omega_n\} forms a basis for K as a vector space over \mathbb{Q}.^[30] The ring of integers \mathcal{O}_K is a free \mathbb{Z}-module of rank n, and thus every number field admits an integral basis.^[31] This existence is established by selecting a \mathbb{Z}-basis of algebraic integers that minimizes the absolute value of the associated discriminant and showing that any non-integral element would allow a refinement yielding a smaller discriminant, leading to a contradiction.^[31] A key property is that if \{\omega_1, \dots, \omega_n\} \subset \mathcal{O}_K spans K over \mathbb{Q}, then the \mathbb{Z}-module M = \sum_{i=1}^n \mathbb{Z} \omega_i satisfies [\mathcal{O}_K : M] < \infty.^[32] The index equals the square root of the ratio of the discriminants of M and \mathcal{O}_K, which is finite since both are nonzero integers.^[32] To compute an integral basis, Minkowski's geometry of numbers embeds \mathcal{O}_K as a full-rank lattice in \mathbb{R}^n via the real and complex embeddings of K, allowing bounds on the successive minima of the lattice to identify short vectors that generate \mathcal{O}_K as a \mathbb{Z}-module.^[32] These bounds limit the coefficients needed to enlarge a finite-index order, such as \mathbb{Z}[\alpha] for a primitive element \alpha \in \mathcal{O}_K, until the full ring is obtained.^[32] For the quadratic field K = \mathbb{Q}(\sqrt{d}) with square-free integer d > 0, an integral basis is \{1, \sqrt{d}\} if d \equiv 2, 3 \pmod{4}, and \{1, (1 + \sqrt{d})/2\} if d \equiv 1 \pmod{4}.^[33] Similar explicit bases exist for imaginary quadratic fields, adjusted for the congruence class of d < 0.^[33]

Power Basis

In an algebraic number field K of degree n over \mathbb{Q}, a power basis for the ring of integers \mathcal{O}_K is a \mathbb{Z}-basis of the form \{1, \alpha, \alpha^2, \dots, \alpha^{n-1}\} for some \alpha \in \mathcal{O}_K such that \mathbb{Z}[\alpha] = \mathcal{O}_K.^[34] Such a ring of integers is called monogenic, meaning it is generated as a \mathbb{Z}-algebra by a single element.^[35] A number field K = \mathbb{Q}(\alpha) is monogenic if the discriminant of the minimal polynomial of \alpha equals the discriminant of K, in which case the powers of \alpha form a power basis for \mathcal{O}_K.^[34] This relation arises because the discriminant of the order \mathbb{Z}[\alpha] is the index [\mathcal{O}_K : \mathbb{Z}[\alpha]]^2 times the field discriminant, so monogenity holds precisely when this index is 1.^[36] All quadratic number fields are monogenic; for example, if d \equiv 2, 3 \pmod{4}, then \mathcal{O}_K = \mathbb{Z}[\sqrt{d}], while if d \equiv 1 \pmod{4}, then \mathcal{O}_K = \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right].^[37] In contrast, not all higher-degree fields are monogenic; a classic counterexample is Dedekind's cubic field K = \mathbb{Q}(\theta), where \theta satisfies the minimal polynomial x^3 + x^2 - 2x - 1 = 0, and \mathcal{O}_K has index greater than 1 over any \mathbb{Z}[\alpha] with \alpha generating K. This example demonstrates that monogenity fails even for fields with class number 1, and counterexamples exist independently of the class number being greater than 1.^[38] A power basis is a special case of an integral basis where the basis elements are powers of a single generator.^[39]

Trace, Norm, and Discriminant

Regular Representation

In an algebraic number field K of degree n = [K : \mathbb{Q}], the regular representation provides a way to view elements of K as linear transformations on the \mathbb{Q}-vector space K itself. Specifically, for any \alpha \in K, the map m_\alpha : K \to K defined by m_\alpha(\beta) = \alpha \beta is a \mathbb{Q}-linear endomorphism of K.^[40] Fix a \mathbb{Q}-basis \{\omega_1, \dots, \omega_n\} for K. The matrix of m_\alpha with respect to this basis, denoted A_\alpha, has entries determined by expressing \alpha \omega_j = \sum_{i=1}^n (A_\alpha)_{i j} \omega_i for each j = 1, \dots, n. The map \alpha \mapsto A_\alpha is a ring homomorphism from K to the ring M_n(\mathbb{Q}) of n \times n matrices over \mathbb{Q}, thereby embedding K as a subring of M_n(\mathbb{Q}).^[40] (S. Lang, Algebra, 3rd ed., Addison-Wesley, 2002, Ch. V, §7) The characteristic polynomial of A_\alpha is \det(x I_n - A_\alpha), which equals the minimal polynomial of \alpha over \mathbb{Q} when \alpha generates K as a \mathbb{Q}-algebra, i.e., when \{1, \alpha, \dots, \alpha^{n-1}\} forms a basis for K. In general, this characteristic polynomial factors as the product \prod_{\sigma} (x - \sigma(\alpha)), where the product runs over all n distinct embeddings \sigma : K \hookrightarrow \mathbb{C}. The trace of A_\alpha equals the sum of the images \sigma(\alpha) under these embeddings.^[40] When K = \mathbb{Q}(\alpha) for some \alpha \in K, the power basis \{1, \alpha, \dots, \alpha^{n-1}\} yields an explicit form for A_\alpha: it is the companion matrix of the minimal polynomial p_\alpha(x) of \alpha over \mathbb{Q}. If p_\alpha(x) = x^n + c_{n-1} x^{n-1} + \dots + c_0, then

A_\alpha = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}.

The characteristic polynomial of this companion matrix is precisely p_\alpha(x).^[40]

Trace and Norm Functions

In an algebraic number field K of degree n = [K : \mathbb{Q}], the trace and norm functions provide key scalar invariants for elements of K, arising from its regular representation as a vector space over \mathbb{Q}. These maps extract the trace and determinant of the linear transformation given by multiplication by an element \alpha \in K on K.^[40] The trace \operatorname{Tr}_{K/\mathbb{Q}} : K \to \mathbb{Q} of \alpha \in K is defined as the sum of the images of \alpha under all n distinct embeddings \sigma : K \hookrightarrow \mathbb{C}:

\operatorname{Tr}_{K/\mathbb{Q}}(\alpha) = \sum_{\sigma} \sigma(\alpha).

^[41] Equivalently, if \{\omega_1, \dots, \omega_n\} is a \mathbb{Q}-basis for K and A_\alpha is the matrix of multiplication-by-\alpha with respect to this basis, then \operatorname{Tr}_{K/\mathbb{Q}}(\alpha) = \operatorname{tr}(A_\alpha), the trace of this matrix.^[40] The trace map is \mathbb{Q}-linear, meaning \operatorname{Tr}_{K/\mathbb{Q}}(c \alpha + \beta) = c \operatorname{Tr}_{K/\mathbb{Q}}(\alpha) + \operatorname{Tr}_{K/\mathbb{Q}}(\beta) for c \in \mathbb{Q} and \alpha, \beta \in K.^[40] For c \in \mathbb{Q}, it simplifies to \operatorname{Tr}_{K/\mathbb{Q}}(c) = n c.^[40] The norm N_{K/\mathbb{Q}} : K \to \mathbb{Q} of \alpha \in K is the product of its images under the embeddings:

N_{K/\mathbb{Q}}(\alpha) = \prod_{\sigma} \sigma(\alpha),

^[41] or equivalently, N_{K/\mathbb{Q}}(\alpha) = \det(A_\alpha), the determinant of the multiplication matrix.^[40] The norm is multiplicative, satisfying N_{K/\mathbb{Q}}(\alpha \beta) = N_{K/\mathbb{Q}}(\alpha) N_{K/\mathbb{Q}}(\beta) for all \alpha, \beta \in K, and maps units to units: N_{K/\mathbb{Q}}(K^\times) \subseteq \mathbb{Q}^\times.^[40] For c \in \mathbb{Q}, N_{K/\mathbb{Q}}(c) = c^n.^[40] When \alpha is an algebraic integer, i.e., \alpha \in \mathcal{O}_K, the minimal polynomial of \alpha over \mathbb{Q} is monic with integer coefficients, implying that both the trace and norm lie in \mathbb{Z}: \operatorname{Tr}_{K/\mathbb{Q}}(\alpha) \in \mathbb{Z} and N_{K/\mathbb{Q}}(\alpha) \in \mathbb{Z}.^[42] The norm extends naturally to nonzero ideals of \mathcal{O}_K: for \mathfrak{a} \subseteq \mathcal{O}_K, N(\mathfrak{a}) = |\mathcal{O}_K / \mathfrak{a}|, the cardinality of the finite quotient ring, which is a positive integer.^[43] This ideal norm is multiplicative over ideal multiplication and coincides with the field norm on principal ideals generated by algebraic integers.^[43] The trace further defines a symmetric bilinear form on K, known as the trace form or trace pairing, given by \langle \alpha, \beta \rangle = \operatorname{Tr}_{K/\mathbb{Q}}(\alpha \beta) for \alpha, \beta \in K.^[44] This form is non-degenerate over \mathbb{Q}, providing a natural inner product structure on the vector space K, with the associated Gram matrix relative to a basis having nonzero determinant.^[44]

Discriminant Properties

The discriminant of an algebraic number field K of degree n over \mathbb{Q} is a fundamental invariant defined using the trace form. Let \mathcal{O}_K be the ring of integers of K, and let \{\omega_1, \dots, \omega_n\} be a \mathbb{Z}-basis for \mathcal{O}_K. The discriminant \operatorname{Disc}_{K/\mathbb{Q}} is given by the determinant

\operatorname{Disc}_{K/\mathbb{Q}} = \det\left( \operatorname{Tr}_{K/\mathbb{Q}}(\omega_i \omega_j) \right)_{1 \leq i,j \leq n},

where \operatorname{Tr}_{K/\mathbb{Q}} denotes the field trace.^[45]^[46] This definition yields an integer that is independent of the choice of \mathbb{Z}-basis for \mathcal{O}_K, as changing the basis by an element of \mathrm{GL}_n(\mathbb{Z}) scales the determinant by the square of the basis change matrix's determinant, which is \pm 1. For primitive elements, if K = \mathbb{Q}(\alpha) with minimal polynomial f(T) \in \mathbb{Z}[T] of degree n and \mathcal{O}_K = \mathbb{Z}[\alpha], then \operatorname{Disc}_{K/\mathbb{Q}} = (-1)^{n(n-1)/2} N_{K/\mathbb{Q}}(f'(\alpha)), where f' is the derivative of f; this is equivalently the resultant \operatorname{Res}(f, f') up to the leading coefficient of f.^[45]^[47]^[46] The discriminant is closely related to the different ideal \mathfrak{d}_K, defined as the inverse of the dual lattice \mathcal{O}_K^\vee = \{\alpha \in K \mid \operatorname{Tr}_{K/\mathbb{Q}}(\alpha \beta) \in \mathbb{Z} \text{ for all } \beta \in \mathcal{O}_K\}. Specifically, \mathfrak{d}_K = (\mathcal{O}_K^\vee)^{-1}, and |\operatorname{Disc}_{K/\mathbb{Q}}| = N_{K/\mathbb{Q}}(\mathfrak{d}_K), where N_{K/\mathbb{Q}} is the absolute norm; thus, the absolute value |\operatorname{Disc}_{K/\mathbb{Q}}| equals the index [\mathcal{O}_K^\vee : \mathcal{O}_K]. This connection highlights the discriminant's role in measuring the "lattice mismatch" between \mathcal{O}_K and its trace dual.^[45]^[47] For quadratic fields K = \mathbb{Q}(\sqrt{d}) with square-free integer d > 0 or d < 0, the discriminant satisfies specific congruence conditions: if d \equiv 2, 3 \pmod{4}, then \operatorname{Disc}_{K/\mathbb{Q}} = 4d \equiv 0 \pmod{4}; if d \equiv 1 \pmod{4}, then \operatorname{Disc}_{K/\mathbb{Q}} = d \equiv 1 \pmod{4}. In general, the sign of the discriminant is (-1)^{r_2}, where r_2 is the number of pairs of complex embeddings, so \operatorname{Disc}_{K/\mathbb{Q}} > 0 precisely when K is totally real (i.e., r_2 = 0). These properties underscore the discriminant's utility as a signature of the field's arithmetic structure.^[45]^[46]^[47]

Places and Valuations

Archimedean Places

Archimedean places of an algebraic number field K, a finite extension of \mathbb{Q} of degree n = [K : \mathbb{Q}], are the infinite places v_\infty arising from the embeddings of K into \mathbb{C}. These places correspond bijectively to the real embeddings \sigma : K \to \mathbb{R} and to the pairs of complex conjugate embeddings \{\sigma, \overline{\sigma}\} : K \to \mathbb{C} (up to conjugation). The number of real embeddings is denoted r_1, and the number of complex conjugate pairs is r_2, yielding the signature (r_1, r_2) of K with r_1 + 2r_2 = n. For example, the field \mathbb{Q}(\sqrt{2}) has signature (2, 0), while \mathbb{Q}(i) has signature (0, 1).^[48] Each Archimedean place v defines an absolute value |\cdot|_v on K. For a real place corresponding to an embedding \sigma : K \to \mathbb{R}, the absolute value is |x|_v = |\sigma(x)|, the standard absolute value on \mathbb{R}. For a complex place corresponding to the pair \{\sigma, \overline{\sigma}\}, it is |x|_v = |\sigma(x)|^2 = |\overline{\sigma}(x)|^2, using the modulus on \mathbb{C} raised to the power 2 to ensure multiplicativity and compatibility with the product formula for absolute values. These valuations are Archimedean, meaning the completion K_v at v is isomorphic to \mathbb{R} for real places or to \mathbb{C} for complex places, both equipped with their standard topologies.^[48] The structure of Archimedean places plays a crucial role in the arithmetic of K, particularly in the geometry of numbers. The group of units \mathcal{O}_K^\times in the ring of integers \mathcal{O}_K is a finitely generated abelian group whose torsion subgroup is the roots of unity in K, and whose free part has rank r_1 + r_2 - 1, as established by Dirichlet's unit theorem. This rank reflects the number of independent units arising from the logarithmic embeddings at the infinite places.^[49]

Non-Archimedean Places

In an algebraic number field K, the non-Archimedean places are in one-to-one correspondence with the nonzero prime ideals \mathfrak{p} of the ring of integers \mathcal{O}_K.^[50] For each such prime ideal \mathfrak{p}, there is a discrete valuation v_{\mathfrak{p}} : K^\times \to \mathbb{Z}, defined as follows: for \alpha \in K^\times, write \alpha = x/y with x, y \in \mathcal{O}_K \setminus \{0\}; then v_{\mathfrak{p}}(\alpha) = \mathrm{ord}_{\mathfrak{p}}(x) - \mathrm{ord}_{\mathfrak{p}}(y), where \mathrm{ord}_{\mathfrak{p}}(z) is the largest nonnegative integer m such that \mathfrak{p}^m divides the principal ideal z \mathcal{O}_K.^[50] This valuation satisfies v_{\mathfrak{p}}(\alpha \beta) = v_{\mathfrak{p}}(\alpha) + v_{\mathfrak{p}}(\beta) and v_{\mathfrak{p}}(1) = 0, with v_{\mathfrak{p}}(0) = +\infty by convention.^[50] The associated non-Archimedean absolute value is the normalized multiplicative function |\cdot|_{\mathfrak{p}} : K \to \mathbb{R}_{\geq 0} given by |\alpha|_{\mathfrak{p}} = N(\mathfrak{p})^{-v_{\mathfrak{p}}(\alpha)} for \alpha \in K^\times, where N(\mathfrak{p}) = |\mathcal{O}_K / \mathfrak{p}| = p^f is the norm of \mathfrak{p}, with p the unique rational prime below \mathfrak{p} and f = [\mathcal{O}_K / \mathfrak{p} : \mathbb{F}_p] the inertial degree.^[50] This absolute value satisfies |\alpha \beta|_{\mathfrak{p}} = |\alpha|_{\mathfrak{p}} \cdot |\beta|_{\mathfrak{p}} and |1|_{\mathfrak{p}} = 1, and it is non-Archimedean in the sense that it obeys the ultrametric inequality: |x + y|_{\mathfrak{p}} \leq \max\{ |x|_{\mathfrak{p}}, |y|_{\mathfrak{p}} \} for all x, y \in K.^[50] Equivalently, v_{\mathfrak{p}}(x + y) \geq \min\{ v_{\mathfrak{p}}(x), v_{\mathfrak{p}}(y) \}.^[50] The completion K_{\mathfrak{p}} of K with respect to the metric d(\alpha, \beta) = |\alpha - \beta|_{\mathfrak{p}} is a complete non-Archimedean valued field that is a finite extension of the p-adic numbers \mathbb{Q}_p, with degree equal to the local degree [\mathcal{O}_K / \mathfrak{p} : \mathbb{F}_p] \cdot e(\mathfrak{p}/p), where e(\mathfrak{p}/p) is the ramification index.^[51] The ring of integers of K_{\mathfrak{p}} is the closure of \mathcal{O}_K in this completion, and the valuation extends uniquely to K_{\mathfrak{p}}.^[51] A fundamental property of these absolute values across all places (non-Archimedean and Archimedean) is the product formula: for any nonzero \alpha \in K^\times,

\prod_v |\alpha|_v = 1,

where the product runs over all places v of K, with finite places normalized as |\alpha|_{\mathfrak{p}} = N(\mathfrak{p})^{-v_{\mathfrak{p}}(\alpha)} and Archimedean places as |\alpha|_v = |\sigma(\alpha)| for real embeddings and |\alpha|_v = |\sigma(\alpha)|^2 for complex places.^[50]^[51] This formula underscores the balance between local and global behavior in number fields.

Prime Ideals and Lying Over

In the ring of integers \mathcal{O}_K of an algebraic number field K of degree n = [K : \mathbb{Q}], the nonzero prime ideals \mathfrak{p} are precisely the maximal ideals, and each such \mathfrak{p} lies above a unique rational prime p, meaning \mathfrak{p} \cap \mathbb{Z} = p\mathbb{Z}, with the norm N(\mathfrak{p}) = p^f where f = [\mathcal{O}_K / \mathfrak{p} : \mathbb{Z}/p\mathbb{Z}] is the residue field degree.^[52] These prime ideals define the non-Archimedean places of K, where the valuation is given by the \mathfrak{p}-adic valuation.^[53] The lying over theorem states that for every rational prime p, the principal ideal (p) factors uniquely in \mathcal{O}_K as (p) \mathcal{O}_K = \prod_{i=1}^g \mathfrak{p}_i^{e_i}, where the \mathfrak{p}_i are distinct prime ideals of \mathcal{O}_K lying over p (i.e., \mathfrak{p}_i \cap \mathbb{Z} = p\mathbb{Z}), and each e_i \geq 1 is the ramification index.^[53] There are g such prime ideals \mathfrak{p}_i above p, and the fundamental decomposition law asserts that \sum_{i=1}^g e_i f_i = n, where f_i = [\mathcal{O}_K / \mathfrak{p}_i : \mathbb{Z}/p\mathbb{Z}] is the residue degree for each i.^[52] In the special case where K/\mathbb{Q} is Galois, all ramification indices e_i are equal to some e, all residue degrees f_i are equal to some f, and thus g = n / (e f).^[54] For a Galois extension L/K of number fields, the decomposition and inertia groups provide a group-theoretic description of this factorization at each prime. Specifically, for a prime \mathfrak{P} of \mathcal{O}_L lying over a prime \mathfrak{p} of \mathcal{O}_K, the decomposition group D_\mathfrak{P} is the stabilizer subgroup \{\sigma \in \mathrm{Gal}(L/K) \mid \sigma(\mathfrak{P}) = \mathfrak{P}\}, which has order e f, while the inertia subgroup I_\mathfrak{P} = \{\sigma \in D_\mathfrak{P} \mid \sigma \equiv \mathrm{id} \pmod{\mathfrak{P}}\} has order e and acts on the residue field extension \mathcal{O}_L / \mathfrak{P} over \mathcal{O}_K / \mathfrak{p}.^[54] The quotient D_\mathfrak{P} / I_\mathfrak{P} is cyclic of order f, generated by the Frobenius element, which describes the Galois action on the residue fields.^[54] A concrete example occurs in quadratic fields K = \mathbb{Q}(\sqrt{d}) for square-free integer d > 0 or d < 0. For an odd prime p not dividing $2d, the prime p splits completely into two distinct prime ideals if the Legendre symbol (d/p) = 1, remains inert (i.e., g=1, e=1, f=2) if (d/p) = -1, corresponding to whether d is a quadratic residue modulo p.^[55]

Ramification Theory

Ramification Indices

In the context of an algebraic number field extension K/\mathbb{Q}, the ramification index e(\mathfrak{p}/p) for a prime ideal \mathfrak{p} of the ring of integers \mathcal{O}_K lying over a rational prime p is defined as the exponent of \mathfrak{p} in the prime ideal factorization of the extended ideal (p)\mathcal{O}_K = \prod_i \mathfrak{p}_i^{e_i}, where the product runs over the distinct prime ideals \mathfrak{p}_i above p. Equivalently, e(\mathfrak{p}/p) = v_{\mathfrak{p}}(p), the \mathfrak{p}-adic valuation of p. This index quantifies the extent to which p ramifies in the extension at \mathfrak{p}, with e(\mathfrak{p}/p) = 1 indicating unramified behavior and e(\mathfrak{p}/p) > 1 indicating ramification.^[56]^[57] The ramification is classified as tame or wild based on the relationship between e(\mathfrak{p}/p) and the characteristic of the residue field \mathbb{F}_p, which is p. Tame ramification occurs when p does not divide e(\mathfrak{p}/p), i.e., \gcd(e(\mathfrak{p}/p), p) = 1; in such cases, the ramification group beyond the inertia subgroup is often trivial, and e(\mathfrak{p}/p) typically divides p^f - 1 where f is the residue field degree. Wild ramification arises when p divides e(\mathfrak{p}/p), leading to more intricate structure involving higher powers of p in the ramification; this regime requires advanced tools such as higher ramification groups and Galois cohomology to describe the filtration of the Galois group.^[57]^[58] A special case is total ramification, where the extension is totally ramified at \mathfrak{p} over p if there is only one prime \mathfrak{p} above p (so the number of primes g = 1), the residue degree f(\mathfrak{p}/p) = 1, and thus e(\mathfrak{p}/p) = [K : \mathbb{Q}] by the formula efg = [K : \mathbb{Q}]. In Galois extensions, the inertia subgroup I_{\mathfrak{p}} of the decomposition group \mathrm{Gal}(K_{\mathfrak{p}} / \mathbb{Q}_p) (where K_{\mathfrak{p}} is the completion at \mathfrak{p}) has order e(\mathfrak{p}/p), consisting of those elements acting trivially on the residue field \mathcal{O}_K / \mathfrak{p}; the full decomposition group has order e(\mathfrak{p}/p) \cdot f(\mathfrak{p}/p).^[56]^[59]

Dedekind Discriminant Theorem

The Dedekind discriminant theorem provides a precise criterion for ramification in terms of the discriminant \Delta_K of an algebraic number field K/\mathbb{Q}: a prime p ramifies if and only if v_p(\Delta_K) > 0. For tame ramification at p, where p does not divide any e(\mathfrak{p}/p), the p-adic valuation is given by

v_p(\Delta_K) = \sum_{\mathfrak{p} \mid p} f(\mathfrak{p}/p) \left( e(\mathfrak{p}/p) - 1 \right).

In the general case, including wild ramification, the valuation is v_p(\Delta_K) = \sum_{\mathfrak{p} \mid p} f(\mathfrak{p}/p) \, v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}), where \mathfrak{D}_{K/\mathbb{Q}} is the different ideal, and v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}) \geq e(\mathfrak{p}/p) - 1, with equality if and only if the ramification is tame at \mathfrak{p}. In the Galois case, v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}) = \sum_{i \geq 0} \frac{|G_i| - 1}{|G_0|}, where G_i are the higher ramification groups of the decomposition group.^[18]^[60] The proof of the theorem relies on the relationship between the discriminant and the different ideal \mathfrak{D}_{K/\mathbb{Q}}, which is the inverse of the trace dual of \mathcal{O}_K with respect to the trace form \operatorname{Tr}_{K/\mathbb{Q}}. The absolute discriminant satisfies \Delta_K = N_{\mathbb{Q}(\mathfrak{D}_{K/\mathbb{Q}})}(\mathfrak{D}_{K/\mathbb{Q}}), the norm of the different ideal from K to \mathbb{Q}. To compute v_p(\Delta_K), one evaluates the local valuations v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}) using the structure of the completion at \mathfrak{p}. The valuation v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}) is determined by the higher ramification groups G_i of the Galois closure, specifically v_{\mathfrak{p}}(\mathfrak{D}_{K/\mathbb{Q}}) = \sum_{i \geq 0} \frac{|G_i| - 1}{|G_0|}, but in the Dedekind formulation, this reduces to the explicit expression involving e and f via the trace form on a basis adapted to the ramification filtration. The trace form's non-degeneracy ensures the discriminant is nonzero, and the local computations yield the summed contributions over primes above p.^[18]^[60] A key implication of the theorem is that \Delta_K \neq 0 for any number field K \neq \mathbb{Q}, as the formula shows positive valuation only at finitely many ramified primes, ensuring the product over all p defines a nonzero integer. Moreover, the primes p dividing \Delta_K are precisely those that ramify in K, since v_p(\Delta_K) > 0 if and only if some e(\mathfrak{p}/p) > 1. This criterion resolves earlier difficulties in Kummer's approach to Fermat's Last Theorem for regular primes by providing a clean arithmetic invariant for detecting ramification.^[61]^[18] Historically, the theorem represents a cornerstone of Dedekind's development of ideal theory in the 1870s, announced in 1871 and fully elaborated in his supplements to Dirichlet's Vorlesungen über Zahlentheorie. Dedekind introduced the discriminant and different ideals to overcome limitations in Kummer's ideal numbers, particularly for handling ramification in cyclotomic fields and irregular primes, thereby unifying the arithmetic of number fields under the framework of Dedekind domains.^[61]^[62]

Ramification Examples

In the quadratic field K = \mathbb{Q}(\sqrt{-3}), the ring of integers is \mathcal{O}_K = \mathbb{Z}[\omega] where \omega = (-1 + \sqrt{-3})/2, and the field discriminant is -3.^[61] The prime ideal (3) ramifies as (3) = \mathfrak{p}^2 where \mathfrak{p} = (\omega), yielding ramification index e = 2, since 3 divides the discriminant to the first power.^[61] For the p-th cyclotomic field K = \mathbb{Q}(\zeta_p) where p is an odd prime and \zeta_p is a primitive p-th root of unity, the ring of integers is \mathcal{O}_K = \mathbb{Z}[\zeta_p], and the extension degree is p-1.^[63] The prime p ramifies totally with ramification index e = p-1 and residue degree f = 1, as (p) = (1 - \zeta_p)^{p-1}, where $1 - \zeta_p generates the unique prime ideal above p with norm p.^[63] For distinct odd primes q \neq p, q remains unramified (e = 1) and either stays inert (f = p-1, g = 1) if \left( \frac{q}{p} \right) = -1, or splits into g = (p-1)/f prime ideals each with residue degree f (the multiplicative order of q modulo p) if \left( \frac{q}{p} \right) = 1.^[63] Consider the cubic field

K = \mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})

, where the ring of integers is

\mathcal{O}_K = \mathbb{Z}[\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2}]

and the field discriminant is -108 = -2^2 \cdot 3^3.^[12] The prime 2 ramifies totally with index e = 3 and f = 1, as

(2) = (\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})^3

.^[12] This ramification is tame, since \gcd(2,3)=1.^[12] To compute ramification indices, residue degrees, and number of primes g for a prime p in a Galois extension K = \mathbb{Q}(\alpha) of degree n, factor the minimal polynomial f(x) of \alpha over \mathbb{Z} modulo p into distinct irreducible factors \overline{f}_i(x) of degrees f_i in \mathbb{F}_p; then p factors into g prime ideals with indices e_i and residue degrees f_i satisfying e_1 f_1 + \cdots + e_g f_g = n and g = number of factors, where the e_i are equal in the Galois case.^[64] For totally ramified primes, the minimal polynomial is Eisenstein at p, ensuring (p) = \mathfrak{p}^n with e = n and f = 1, as in the examples above for 2 in

\mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})

via x^3 - 2.^[64] These factorizations verify ramification via the Dedekind discriminant theorem.^[64] Ramification is tame if the residue characteristic p does not divide the index e, and wild otherwise.^[65] In quadratic fields K = \mathbb{Q}(\sqrt{d}) with d < 0, ramification at 2 (when it occurs) is typically wild, since e = 2 is divisible by p = 2.^[65] For instance, in \mathbb{Q}(\sqrt{-6}), 2 ramifies wildly with e = 2.^[65]

Galois Extensions

Galois Groups

In the context of an algebraic number field K of degree n = [K : \mathbb{Q}], the Galois group is typically considered with respect to the Galois closure L of K over \mathbb{Q}, which is the smallest Galois extension of \mathbb{Q} containing K. This closure L is obtained by adjoining all conjugates of a primitive element for K to \mathbb{Q}, and the Galois group \mathrm{Gal}(L / \mathbb{Q}) consists of all \mathbb{Q}-automorphisms of L, acting faithfully on the roots of the minimal polynomial of that primitive element by permuting them.^[66] The action extends to the embeddings of K into \overline{\mathbb{Q}}, the algebraic closure of \mathbb{Q}, where \mathrm{Gal}(L / \mathbb{Q}) permutes these embeddings transitively if K is generated by a single element.^[67] The absolute Galois group of \mathbb{Q}, denoted \mathrm{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}), is the inverse limit of the Galois groups of all finite Galois extensions of \mathbb{Q}, equipped with the Krull topology, and it acts on the roots of unity and other algebraic elements in \overline{\mathbb{Q}}.^[66] For a general number field K, the relevant Galois group is thus a quotient of this absolute group, specifically the image of \mathrm{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) in the permutations of the roots defining L.^[67] When K / \mathbb{Q} is itself Galois, the Galois group simplifies to \mathrm{Gal}(K / \mathbb{Q}), which is isomorphic to the automorphism group \mathrm{Aut}_{\mathbb{Q}}(K) of field automorphisms fixing \mathbb{Q}, and its order equals the degree n = [K : \mathbb{Q}].^[66] In this case, the fundamental theorem of Galois theory establishes a bijection between the subgroups of \mathrm{Gal}(K / \mathbb{Q}) and the intermediate fields between \mathbb{Q} and K: for a subgroup H \leq \mathrm{Gal}(K / \mathbb{Q}), the fixed field K^H = \{ x \in K \mid \sigma(x) = x \ \forall \sigma \in H \} is a subextension, and conversely, every subfield arises this way, with the correspondence reversing inclusion and preserving indices.^[66] A basic example is the quadratic field K = \mathbb{Q}(\sqrt{d}) for square-free integer d > 0, which is Galois over \mathbb{Q} with \mathrm{Gal}(K / \mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}, generated by the automorphism sending \sqrt{d} to -\sqrt{d}.^[67] For cyclotomic fields K = \mathbb{Q}(\zeta_m), where \zeta_m is a primitive m-th root of unity, the extension is Galois over \mathbb{Q} with \mathrm{Gal}(K / \mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^\times, the multiplicative group of integers modulo m coprime to m, of order \varphi(m) (Euler's totient function); the isomorphism sends a \in (\mathbb{Z}/m\mathbb{Z})^\times to the automorphism \sigma_a defined by \sigma_a(\zeta_m) = \zeta_m^a.^[68] In a Galois extension L / K of number fields, for an unramified prime \mathfrak{p} of K, the decomposition group D_{\mathfrak{q}} at a prime \mathfrak{q} of L above \mathfrak{p} is the stabilizer subgroup of \mathfrak{q} in \mathrm{Gal}(L / K), and it is cyclic of order equal to the residue degree f(\mathfrak{q} / \mathfrak{p}). The Frobenius element \mathrm{Frob}_{\mathfrak{q}} \in D_{\mathfrak{q}} is the unique generator of this group that acts on the residue field extension by raising elements to the power N(\mathfrak{p}) (the norm of \mathfrak{p}), satisfying \sigma(x) \equiv x^{N(\mathfrak{p})} \pmod{\mathfrak{q}} for x \in \mathcal{O}_L. All such Frobenius elements for primes above \mathfrak{p} form a conjugacy class in \mathrm{Gal}(L / K).^[54]

Galois Cohomology Applications

Galois cohomology provides a powerful framework for studying the arithmetic of algebraic number fields K by computing invariants of the absolute Galois group G_K = \Gal(K^s / K), where K^s denotes the separable closure of K. For a discrete G_K-module M, the Galois cohomology groups are defined as H^i(G_K, M) = \varinjlim H^i(\Gal(L/K), M^{\Gal(K^s/L)}), where the direct limit is taken over all finite Galois extensions L/K.^[69] These groups capture obstructions and extensions in Galois theory, with applications to arithmetic structures like units and ideals.^[70] A key application arises in the study of the ideal class group modulo \ell, for a prime \ell. For suitable finite extensions, the group

H^1(\Gal(L/K), K^s^\times)

relates to the relative norm index, but more precisely, the \ell-primary component of the class group \Cl(K) can be analyzed via cohomology with coefficients in \ell-torsion modules, yielding isomorphisms such as H^1(G_K, \mu_\ell) \cong K^\times / (K^\times)^\ell under the assumption that \mu_\ell \subset K, which connects to the \ell-structure of ray class groups.^[69] Another central application is to the Brauer group \Br(K), which classifies central simple algebras over K up to Morita equivalence and is isomorphic to

H^2(G_K, K^s^\times)

. For number fields, \Br(K) injects into the direct sum of local Brauer groups, with the kernel measuring global obstructions.^[70]^[69] Kummer theory exemplifies these ideas by classifying cyclic extensions of exponent n when K contains the nth roots of unity \mu_n. The Kummer map K^\times / (K^\times)^n \to H^1(G_K, \mu_n) is an isomorphism, parametrizing such extensions by elements of K^\times modulo nth powers; the kernel corresponds to norms from the extension.^[69] For example, adjoining an nth root of an element a \in K^\times yields a cyclic extension if and only if the image of a in the cohomology group generates a cyclic subgroup.^[60] Tate cohomology extends ordinary cohomology to negative degrees for finite Galois groups, providing invariants like the normalized groups \hat{H}^i(G, M) for i \in \mathbb{Z}. In the context of a finite Galois extension L/K of number fields with group G = \Gal(L/K), \hat{H}^0(G, \mathcal{O}_L^\times) \cong \mathcal{O}_K^\times identifies the global units as G-invariants, while \hat{H}^{-1}(G, I_L) \cong \Cl(\mathcal{O}_K) links the relative class group to the cohomology of the group of fractional ideals I_L. These isomorphisms preview the structure theorems of class field theory, where the full idele class group replaces I_L.^[69]^[60] The Herbrand quotient further refines these computations, defined for a finite G-module M of exponent prime to |G| as h_G(M) = \# H^0(G, M) / \# H^1(G, M), which equals the degree [L:K] for modules like the idele class group in number fields. For abelian extensions, this quotient governs the equality of cohomology dimensions and implies finiteness results, such as the vanishing of certain higher cohomology groups in global fields.^[60]^[70]

Local-Global Principles

Local and Global Fields

In algebraic number theory, a global field in the context of number fields is a finite extension K/\mathbb{Q}.^[51] Such fields possess a rich arithmetic structure arising from their infinitely many places, which are equivalence classes of nontrivial absolute values on K.^[51] A local field associated to a global number field K is the completion K_v of K with respect to a place v.^[71] For a finite (non-archimedean) place v lying over a prime p of \mathbb{Z}, K_v is a finite extension of the p-adic field \mathbb{Q}_p; for an infinite (archimedean) place, K_v is isomorphic to either \mathbb{R} or \mathbb{C}.^[71] Non-archimedean local fields are complete with respect to a discrete valuation and form complete discrete valuation rings (DVRs) with finite residue fields.^[71] Archimedean local fields \mathbb{R} and \mathbb{C} are uniquely determined up to isomorphism as the only such fields of their respective degrees over \mathbb{Q}.^[71] Non-archimedean local fields, being finite extensions of \mathbb{Q}_p, admit unramified extensions of any given degree that are unique up to isomorphism.^[71] A fundamental distinction between global and local fields lies in their valuation structures: while a global field K has infinitely many places—finitely many archimedean and infinitely many non-archimedean—a local field K_v possesses a single uniformizer \pi_v generating its maximal ideal.^[51] This uniformity simplifies analysis in the local setting, enabling tools like Hensel's lemma. Hensel's lemma provides a mechanism for lifting solutions of polynomial equations from the residue field to the local field.^[72] Specifically, in a non-archimedean local field with complete DVR \mathcal{O}_v and maximal ideal \mathfrak{p}_v, if a monic polynomial f \in \mathcal{O}_v has a simple root \bar{r} modulo \mathfrak{p}_v (i.e., f(\bar{r}) \equiv 0 \pmod{\mathfrak{p}_v} and f'(\bar{r}) \not\equiv 0 \pmod{\mathfrak{p}_v}), then there exists a unique root r \in \mathcal{O}_v lifting \bar{r}.^[72] The proof constructs this lift iteratively using Newton's method, converging due to the completeness and the non-archimedean valuation satisfying |f(a_n)| < |f'(a_n)|^2 at each step.^[72] This lemma is pivotal for solving equations over local fields and underlies much of the local analysis in global arithmetic problems.^[72]

Hasse Principle

The Hasse principle, or local-global principle, asserts that a quadratic form q over an algebraic number field K is isotropic—that is, it admits a non-trivial zero in K—if and only if it is isotropic over the completion K_v at every place v of K.^[73] This equivalence reduces the global solvability problem to checking local conditions at finitely many archimedean places and infinitely many non-archimedean places, where local solvability over K_v is often more tractable due to properties of local fields.^[73] For quadratic forms, the Hasse principle holds by the Hasse-Minkowski theorem, which confirms that isotropy over K is equivalent to local isotropy everywhere; this result generalizes Minkowski's theorem over the rationals to arbitrary number fields. The theorem provides a complete classification of quadratic forms up to equivalence via local invariants, such as the Hasse invariant at each place. Despite its success for quadratics, the Hasse principle fails for varieties of higher degree. A seminal counterexample is Selmer's plane cubic curve over \mathbb{Q} given by $3x^3 + 4y^3 + 5z^3 = 0, which possesses non-trivial points over \mathbb{R} and every \mathbb{Q}_p but no non-trivial rational points.^[74] Such failures highlight limitations of local information for predicting global solutions in Diophantine equations. Some of these counterexamples are accounted for by the Brauer-Manin obstruction, which detects violations using the Brauer group \mathrm{Br}(X) of the variety X via a pairing that obstructs rational points when the Brauer-Manin set is empty, even if local points exist everywhere. The principle originated with Helmut Hasse's work in the 1920s, building on p-adic methods to establish local-global equivalences for quadratic forms.^[75] Counterexamples like Selmer's appeared in the 1950s, prompting deeper investigations into obstructions beyond local solubility.^[74]

Adeles and Ideles

In the context of an algebraic number field K, the adele ring \mathfrak{A}_K is defined as the restricted direct product \prod_v' K_v over all places v of K, where K_v denotes the completion of K at v, and the restriction requires that for all but finitely many finite places v, the component lies in the valuation ring \mathcal{O}_{K_v}.^[76] This construction equips \mathfrak{A}_K with a topology making it a locally compact topological ring, where open sets are generated by products of open sets in each K_v such that for almost all finite v, the sets are neighborhoods of the identity in \mathcal{O}_{K_v}.^[76] The diagonal embedding K \hookrightarrow \mathfrak{A}_K, sending x \in K to the tuple (x_v)_v with x_v = x in each K_v, is dense in \mathfrak{A}_K with respect to this topology.^[76] The idele group \mathfrak{A}_K^\times, or ideles of K, consists of the multiplicative units of the adele ring and is given by the restricted direct product \prod_v' K_v^\times, where for infinite places v, components are arbitrary in K_v^\times, and for finite places v, they lie in \mathcal{O}_{K_v}^\times for all but finitely many such v.^[77] Like the adeles, the ideles inherit a locally compact topology from the restricted product, which is stronger than the subspace topology induced from \mathfrak{A}_K^\times, ensuring that the group of principal ideles K^\times embeds densely via the diagonal map.^[77] The idelic norm on \mathfrak{A}_K^\times is defined as the map to \mathbb{R}^\times_{>0} given by the product of local norms N_{K_v/\mathbb{Q}_p}(x_v) (or absolute values at infinite places), and the regulator of K arises as the volume of the image of the unit group \mathcal{O}_K^\times in the idele class group \mathfrak{A}_K^\times / K^\times with respect to the Haar measure normalized by the product formula.^[77] A fundamental property is the product formula for the idelic Haar measure: for x \in \mathfrak{A}_K^\times, the measure satisfies \prod_v |N_{K_v/\mathbb{Q}_p}(x_v)| = 1 when x is in the image of K^\times, which extends the classical product formula for nonzero elements of K and ensures the quotient \mathfrak{A}_K / K is compact.^[76] In class field theory, the idele class group \mathfrak{A}_K^\times / K^\times forms the basis for the class formation, where abelian extensions of K correspond to open subgroups of finite index via Artin reciprocity, linking global class groups to local unit groups.^[77] Adeles and ideles also play a central role in the study of Tamagawa numbers for algebraic groups over K. For a connected reductive algebraic group G defined over K, the Tamagawa number \tau(G) is the volume of the adelic quotient G(K) \backslash G(\mathfrak{A}_K) with respect to the canonical Tamagawa measure on G(\mathfrak{A}_K), a left-invariant Haar measure constructed from invariant differential forms, which equals 1 for simply connected semisimple groups over number fields (as proved by Kottwitz, resolving Weil's conjecture on Tamagawa numbers).^[78]^[79] This measure leverages the restricted product structure to normalize local volumes, providing a global invariant that refines local-global principles in arithmetic geometry.^[78]

References

[1]
[PDF] Algebraic Number Theory - James Milne
Sep 28, 2008 · An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number ...
[2]
Number Field -- from Wolfram MathWorld
The totality of all expressions that can be constructed from r by repeated additions, subtractions, multiplications, and divisions is called a number field.
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Algebraic Number Field - an overview | ScienceDirect Topics
An algebraic number field is defined as a field extension of the rational numbers that is generated by the roots of a polynomial with integer coefficients.
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[PDF] Algebraic Numbers and Algebraic Integers
Definition 1.3. The minimal polynomial f of an algebraic number α is the monic polynomial in Q[X] of smallest degree such that f(α) ...<|separator|>
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[PDF] SEPARABILITY 1. Introduction Let K be a field. We are going to look ...
Definition 1.1. A nonzero polynomial f(X) ∈ K[X] is called separable when it has distinct roots in a splitting field over K. That is, each root of f(X) has ...
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[PDF] Dedekind's 1871 version of the theory of ideals∗ - andrew.cmu.ed
Mar 19, 2004 · [7] Richard Dedekind. Theory of Algebraic Integers. Cambridge University. Press, Cambridge, 1996. A translation of [4], translated and ...
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[PDF] Number Fields and Galois Theory
2 Number Fields. Definition 2.1. Algebraic number fields K, also known as number fields, are finite degree extension fields of Q. In other words, the following ...
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[PDF] Inverse Galois Problem for Totally Real Number Fields
2 (Primitive Element Theorem). If K/F is a finite, separable field extension, then K/F is a simple extension. 5. Page 9. Since Q has characteristic zero, all ...
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[PDF] Algebraic Number Theory Notes - Math (Princeton)
Sep 28, 2025 · Consider now the separable extension L|K given by the primitive element θ ∈ OL with minimal polynomial F(X) ∈ OK[X] so that L = K(θ) and ...
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[PDF] Factoring in quadratic fields - Keith Conrad
Introduction. For a squarefree integer d other than 1, let. K = Q[. √ d] = {x + y. √ d : x, y ∈ Q}. This is called a quadratic field and it has degree 2 over Q.
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[PDF] Quadratic Fields
So when we speak of factorization in Z[√D] being unique, we will always mean unique up to insertion of units (and their inverses). The Norm. We introduce now a ...
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[PDF] cyclotomic extensions - keith conrad
For a field K, an extension of the form K(ζ), where ζ is a root of unity, is called a cyclotomic extension of K. The term cyclotomic means “circle-dividing,” ...
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[PDF] 6.2 Cyclotomic Extensions - UCSB Math
Nov 23, 2021 · Part of why we care about cyclotomic extensions of Q is because of the following theorem. Theorem (Kronecker–Weber). Let K/Q be an abelian ...
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[PDF] the splitting field of x3 − 2 over q - Keith Conrad
minimal polynomial over Q is f(T) = T6 +3T4 +4T3 +3T2 + 1, so the discriminant of Z[δ]. Page 15. THE SPLITTING FIELD OF X3 − 2 OVER Q. 15 is. −NK/Q(f. 0. (δ)) ...Missing: ∛ | Show results with:∛
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[PDF] Applications of Galois theory - Keith Conrad
The extension Q( 3. √. 2,ω)/Q is called non-abelian since its Galois group is isomorphic to S3, which is a non-abelian group. The term “non-abelian” has nothing ...
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[PDF] September 21: Math 432 Class Lecture Notes
The discriminant of a quadratic field is defined to be D = (δ − δ0)2. If d ≡ 2,3 mod 4, then D = 4d, and if d ≡ 1 mod 4 then D = d. There are only ...
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[PDF] Contents - Evan Dummit
A number field with s = 0 is totally real (all its embeddings are real) while a number field with r = 0 is totally complex (all its embeddings are nonreal) ...
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[PDF] CM FIELDS, THE COLMEZ CONJECTURE, AND THETA LIFTING
A totally imaginary number field E is a CM field if E is a quadratic extension of a totally real number field F. Since Q is a totally real number field, any ...
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[PDF] Algebraic Number Theory - James Milne
An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the ...
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An Introduction to the Theory of Field Extensions
Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and ...
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### Summary: Definition and Distinction Between Number Fields and Function Fields
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[PDF] Number Fields - DPMMS
Apr 4, 2019 · In this course we will associate to any number field L a subring OL ⊂ L called the ring of integers of L. This will generalize the inclusion Z ⊂ ...
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[PDF] In this lecture, we continue our discussion of Dedekind domains ...
Oct 5, 2021 · As we shall see in Corollary 2.6, rings of algebraic integers in algebraic number fields are Dedekind domains.
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[PDF] NUMBER RINGS - of /websites - Universiteit Leiden
A number ring is a subring of a number field, which is a finite field extension of the field of rational numbers.
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[PDF] 2. Ideals in Quadratic Number Fields
This chapter introduces the ring of integers in quadratic number fields, modules, and ideals. The ring of integers in K = Q(√m) is denoted as OK.
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[PDF] Kummer's theory on ideal numbers and Fermat's Last Theorem
The failure of unique factorization in the ring of integers of certain cyclotomic fields is what motivated Ernst Kummer to develop his theory of ideal numbers, ...Missing: history | Show results with:history
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[PDF] A brief history of rings - OU Math
Mar 24, 2015 · Kummer invented ideal numbers to figure this out, and maybe to salvage Lam`e's proof. The idea—suppose our ring doesn't have unique ...
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[PDF] ALGEBRAIC NUMBER THEORY Contents Introduction ...
rings of integers in number fields are Dedekind domains, and hence that their ideals factor uniquely into products of prime ideals. Theorem 3.30. Let A be a ...<|control11|><|separator|>
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[PDF] ideal factorization - keith conrad
The ring OK is a unique factorization domain if and only if it is a principal ideal domain. Proof. It is a general theorem of algebra that every PID is a UFD.
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[PDF] Ideal classes and Kronecker bound - Keith Conrad
While the ring of integers of a number field has a finite ideal class group ... For instance, h(Q(. √. −5)) = 2 but h(Q(i,. √. −5)) = 1. It was ...
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### Definition of Integral Basis
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[PDF] Part II - Number Fields - Dexter Chua
The important theorem is that an integral basis always exists. Theorem. Let Q/L be a number field. Then there exists an integral basis for. OL. In particular, ...
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[PDF] Algebraic integers
the ring of integers are more subtle than number fields. Nevertheless, there are computational tools for writing down an integral basis. It is easy to generate ...
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[PDF] Math 210B. Quadratic integer rings 1. Computing the integral ...
reciprocal −1 + √ 2; the general structure of unit groups of rings of integers of number fields is a key part of classical algebraic number theory, beyond the ...Missing: construction | Show results with:construction
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[PDF] A note on families of monogenic number fields.
A number field K|Q is called monogenic if its ring of integers has a power basis,. i.e. is of the form Z + αZ + ··· + αn.
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[PDF] On Power Bases in Number Fields - William Stein
Mar 13, 2005 · A number field K has a power basis (or in some literature, a power integral basis) if its ring of integers is generated by a single element; ...
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[PDF] Rings of integers without a power basis - Keith Conrad
In 1882, Kronecker [7, p. 119] said he found an example in 1858 of a field K in Q(ζ13) where OK has no power basis over Z. It is the quartic subfield K, and it ...
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[PDF] PSLр2; 5Ю SEXTIC FIELDS WITH A POWER BASIS
A field having a power basis is called monogenic. Every quadratic field is monogenic. Dedekind. [5] gave an example of a cubic field which is not monogenic. If ...
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[PDF] 18.785 Number Theory Fall 2021 Problem Set #3 Description
Show that infinitely many cubic number fields are not monogenic and give an example. Problem 4. Orders in Dedekind domains (32 points). Let O be an order ...
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[PDF] Monogenity and Power Integral Bases: Recent Developments - arXiv
Jun 29, 2024 · A non-zero irreducible polynomial f(x) ∈ Z[x] is called monogenic if a root α of f(x) generates a power integral basis in the field K = Q(α).
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[PDF] TRACE AND NORM 1. Introduction Let L/K be a finite extension of ...
By choosing a K-basis of L we can create a matrix representation for mα, which is denoted [mα]. (We need to put an ordering on the basis to get a matrix, but ...
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Introduction - The Trace And The Norm
Let K be a number field. Define the 'trace' T K and the 'norm' N K as follows. Let σ 1 , . . . , σ n be the embeddings of K in C where n = [ K : Q ] .
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[PDF] Contents 0 Algebraic Number Theory - Evan Dummit
0.2 (Sep 5) Rings of Integers, Trace and Norm 1 . ... ◦ Proof: If α is an algebraic integer, its Galois conjugates are also algebraic integers, hence so too are.<|control11|><|separator|>
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[PDF] 6 Ideal norms and the Dedekind-Kummer theorem - MIT Mathematics
Sep 25, 2017 · ... definition of the ideal norm in terms of the field norm. In view of this we extend our definition of the field norm NL/K to fractional ideals.
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[PDF] Do Trace Forms Characterise Number Fields?
Jul 2, 2013 · The discriminant completely characterises a quadratic number field. But for higher degree number fields, this is not the case.
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[PDF] the different ideal - keith conrad
The ideal norm of the different ideal is the absolute value of the discriminant, and this connection between the different and discriminant will tell us ...
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[PDF] 12 The different and the discriminant
Oct 20, 2016 · In this situation it is customary to define the absolute discriminant DL of the number field L to be the integer disc(e1,...,en) ∈ Z, for ...
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[PDF] Math 129: Number Fields
Let p be an odd prime, and consider the cyclotomic field K = Q(ζp), where ζp denotes a primitive p-th root of unity. What is [K ∶ Q]?. Lemma 7.5. The minimal ...
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[PDF] Algebraic Number Theory, a Computational Approach - William Stein
Nov 14, 2012 · J.W.S. Cassels, Global fields, Algebraic Number Theory (Proc. In- structional Conf., Brighton, 1965), Thompson, Washington, D.C.,. 1967, pp ...Missing: regular | Show results with:regular
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[PDF] dirichlet's unit theorem - keith conrad
Introduction. Dirichlet's unit theorem describes the structure of the unit group of orders in a number field. Theorem 1.1 (Dirichlet, 1846).
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[PDF] Keith Conrad - OSTROWSKI FOR NUMBER FIELDS
Ostrowski classified nontrivial absolute values on Q as p-adic and archimedean. This extends to number fields K, with p-adic, real, and complex absolute values.
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[PDF] 13 Global fields and the product formula - MIT Mathematics
Oct 23, 2017 · Up to this point we have defined global fields as finite extensions of Q (number fields) or. Fq(t) (global function fields).
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Summary of each segment:
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[PDF] 6 Ideal norms and the Dedekind-Kummer theorem - MIT Mathematics
Sep 27, 2021 · Theorem 6.10. Assume AKLB and let q be a prime lying above p. Then NB/A(q) = pfq , where fq = [B/q : A/p] is the residue field degree of q.
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[PDF] 5 Factoring primes in Dedekind extensions
Sep 24, 2015 · The primes p of A are all maximal ideals, so each has an associated residue field A/p, and similarly for primes q of B. If q lies above p then ...
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[PDF] 7 Galois extensions, Frobenius elements, and the Artin map
Sep 29, 2021 · The corresponding inertia field LIp and decomposition field LDp are Galois extensions of K that are characterized by the following corollary.
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[PDF] MATH 154. ALGEBRAIC NUMBER THEORY 1. Fermat's ...
= {±1}. (2) For F a field, F. ×. = F − {0} by the definition of a field. Page 3. MATH 154. ALGEBRAIC NUMBER THEORY. 3. (3) We have that 1 +. √. 2 is a unit in Z ...
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[PDF] Primes Presentation - UC Berkeley math
Nov 6, 2018 · In a number field K, OK is the ring of algebraic integers. Every nonzero prime ideal of OK is maximal, and ideals factor uniquely as a product ...
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https://math.mit.edu/classes/18.785/2019fa/LectureNotes11.pdf
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[PDF] The Kronecker-Weber Theorem - UChicago Math
First we recall the definition of tame and wild ramification: Definition. We say that Q/P is tamely ramified if the ramification index e(Q/P) is relatively ...<|separator|>
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[PDF] 11 Totally ramified extensions and Krasner's lemma
In this lecture we will classify totally ramified extensions of complete DVRs. 11.1 Totally ramified extensions of a complete DVR. Definition ...
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[PDF] Algebraic Number Theory - UCSB Math
It is a very sad moment for me to write this "Geleitwort" to the English translation of Jurgen Neukirch's book on Algebraic Number Theory. ... LANG's ...
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[PDF] Discriminants and ramified primes - Keith Conrad
The discriminant of OK is discZ(OK). The next lemma says reduction modulo p commutes (in a suitable sense) with the formation of discriminants. Lemma 3.3.
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https://kconrad.math.uconn.edu/math5230f12/handouts/totram.pdf
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[PDF] Cyclotomic Fields with Applications - G Eric Moorhouse
The symbol ζ denotes a complex root of unity, except when it represents a zeta function (`a la Riemann, Dedekind, Hasse, etc.). Likewise, 'i' signifies either ...
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[PDF] Totally ramified primes and Eisenstein polynomials - Keith Conrad
Let K be a number field, and suppose there is a prime p which is totally ramified in K. Then K = Q(α) for some α which is the root of an Eisenstein polynomial.
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[PDF] Math 676. Tame ramification and composite fields
For counterexamples in the tame and wild cases, we consider F = Qp and non-isomorphic ramified ... For odd p, there are exactly two quadratic ramified extensions, ...
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[PDF] Fields and Galois Theory - James Milne
These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental ...
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[PDF] Galois groups as permutation groups - Keith Conrad
Introduction. A Galois group is a group of field automorphisms under composition. By looking at the effect of a Galois group on field generators we can ...
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[PDF] A Short Course on Galois Cohomology - William Stein
The goal for the rest of the course (about 15 lectures), is to see some applications of group cohomology to Galois theory and algebraic number theory that are.
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[PDF] Galois Cohomology
Jean-Pierre Serre. Galois. Cohomology translated from the French by Patrick Ion. Springer. Page 2. Table of Contents. Foreword. Chapter I. Cohomology of ...
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[PDF] 9 Local fields and Hensel's lemmas
Oct 4, 2017 · In this lecture we introduce the notion of a local field; these are precisely the fields that arise as completions of a global field (finite ...
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[PDF] 18.785 (F2021) Lecture 9: Local Fields and Hensel's Lemmas
Oct 6, 2021 · In this lecture we introduce the notion of a local field; these are precisely the fields that arise as completions of a global field (finite ...
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[PDF] Hasse-Minkowski Theorem - CSUN
Theorem 1.1. (Hasse-Minkowski) Let K be a number field and let q be a quadratic form in n variables with coefficients in K. Then q ...
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The diophantine equation ax3+by3+cz3=0. - Project Euclid
1951 The diophantine equation ax3+by3+cz3=0. Ernst S. Selmer. Author Affiliations +. Ernst S. Selmer1 1Oslo. DOWNLOAD PDF + SAVE TO MY LIBRARY. Acta Math.
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Helmut Hasse - Biography - MacTutor - University of St Andrews
In October 1920 Hasse discovered the 'local-global' principle which shows that a quadratic form that represents 0 non-trivially over the p p p-adic numbers for ...
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https://kskedlaya.org/cft/sec_ideles.html
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6.2 Idèles and class groups - Kiran S. Kedlaya
v . As a set, I K is the restricted product of the pairs ( K v ∗ , { 1 } ) for infinite places v and ( K v , o K v ∗ ) for finite places . v . We use this ...
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[PDF] Weil's Conjecture on Tamagawa Numbers (Lecture 1)
Jan 30, 2014 · ... Tamagawa number of the algebraic group SOq. More generally, if G is a connected semisimple algebraic group over Q, we define the Tamagawa number ...