Fact-checked by Grok 2 weeks ago

Field extension

In abstract algebra, a field extension consists of a base field K and a larger field L such that K is a subfield of L, allowing L to be viewed as a vector space over K.^[1]^[2] The dimension of this vector space, denoted [L:K], is called the degree of the extension and measures its "size" relative to K; extensions with finite degree are termed finite extensions.^[3]^[4] Elements \alpha \in L are classified as algebraic over K if they satisfy a nonzero polynomial equation with coefficients in K, or transcendental otherwise; an extension L/K is algebraic if every element of L is algebraic over K, and transcendental if it contains at least one transcendental element.^[1]^[2] All finite extensions are algebraic, but infinite algebraic extensions exist, such as the field of all algebraic numbers over the rationals \mathbb{Q}.^[3] Simple extensions, generated by adjoining a single element \alpha to K to form K(\alpha), have degree equal to the degree of the minimal polynomial of \alpha over K when \alpha is algebraic.^[2]^[4] Field extensions underpin key results in algebra, including the existence of splitting fields for polynomials—minimal extensions where a given polynomial factors completely into linear terms—and tower laws for degrees in chains of extensions, where [L:K] = [L:F] \cdot [F:K] for intermediate fields.^[1] They are central to Galois theory, which studies the symmetries of extensions via Galois groups to determine solvability of polynomials by radicals, as in classical problems like angle trisection or cube duplication.^[3]^[4] Examples include \mathbb{Q}(\sqrt{2})/\mathbb{Q} (degree 2, algebraic) and \mathbb{R}/\mathbb{Q} (infinite degree, transcendental elements like \pi).^[2]

Definitions and Basic Terminology

Field extensions

A field extension is formally defined as a pair of fields L and K, denoted L/K, where K is a subfield of L.^[5] In this setup, L contains an isomorphic copy of K via an injective field homomorphism, ensuring that the algebraic structure of K is preserved within L.^[3] Common notation for such extensions includes L \supseteq K or K \subseteq L, with the embedding \iota: K \to L typically taken as the identity map when K is literally a subset of L.^[5] This embedding guarantees that the addition and multiplication operations in L restrict exactly to those in K, maintaining compatibility between the fields.^[3] As a basic property, L forms a left vector space over K, where scalar multiplication is defined using the multiplication in L.^[5] The operations of addition and multiplication in L thus extend those of K naturally, allowing elements of K to act as scalars on elements of L. It is important to distinguish a field extension from a mere set inclusion: not every superset of a field qualifies as an extension unless equipped with an embedding that aligns the field operations precisely.^[3] Without this embedding, the structure may fail to preserve the field axioms or compatibility. The concept of field extensions was introduced by Leopold Kronecker in the 19th century, primarily in the context of algebraic number theory, to study rings of algebraic integers within larger fields.^[6]

Subfields

A subfield of a field K is a subset F \subseteq K that forms a field under the addition and multiplication operations induced from K.^[7] Equivalently, a subfield is a subring of K that is itself a field, meaning it contains the multiplicative identity of K and every nonzero element of the subring has a multiplicative inverse within the subring.^[7] Subfields satisfy closure properties inherent to fields: they are closed under addition and multiplication, contain the additive identity 0 and the multiplicative identity 1 of K, and are closed under additive inverses.^[8] Additionally, for every nonzero element a \in F, the multiplicative inverse a^{-1} lies in F.^[8] These properties ensure that the subfield operations align perfectly with those of the ambient field K. The prime subfield of K is the smallest subfield of K, defined as the intersection of all subfields of K.^[9] This prime subfield is unique and isomorphic to \mathbb{[Q](/page/Q)} when the characteristic of K is 0, or to the prime field \mathbb{F}_p when the characteristic of K is a prime p.^[10] Every subfield of K contains the prime subfield, and subfields of K correspond precisely to the subrings of K that are fields under the induced operations.^[7] Intermediate fields arise as subfields strictly between the prime subfield and K, forming chains such as P \subset [F_1](/page/Intermediate) \subset [F_2](/page/Intermediate) \subset \cdots \subset K, where P denotes the prime subfield.^[11] The collection of all subfields of K, ordered by inclusion, constitutes a partially ordered set with the prime subfield as the least element and K as the greatest.^[12]

Degree of a field extension

In field theory, given a field extension L/K where K is a subfield of L, the degree of the extension, denoted [L : K], is defined as the dimension of L considered as a vector space over K.^[13] This dimension measures the "size" of the extension in a linear algebraic sense. Every field extension L/K admits a Hamel basis, which is a linearly independent set over K that spans L as a K-vector space; the existence of such a basis relies on the axiom of choice.^[14] The extension is finite if this basis has finite cardinality n, in which case [L : K] = n, a positive integer; otherwise, the degree is infinite.^[13] Notably, [L : K] = 1 if and only if L = K.^[13] For infinite-degree extensions, the cardinality of a Hamel basis can vary; for instance, the extension \mathbb{Q}(x)/\mathbb{Q} of rational functions has countably infinite degree, while \mathbb{R}/\mathbb{Q} requires an uncountable basis of cardinality equal to the continuum.^[13] In transcendental extensions, the vector space dimension is infinite, and the transcendence degree—defined as the cardinality of a maximal algebraically independent subset over K—provides a measure of the "transcendental part" of the extension, often aligning with the structure of purely transcendental cases like rational function fields.^[15]

Properties of Field Extensions

Simple extensions

A field extension L/K is called a simple extension if there exists some \alpha \in L such that L = K(\alpha), meaning L is generated by adjoining a single element \alpha to K.^[16] This element \alpha is known as a primitive element for the extension L/K.^[17] The elements of a simple extension K(\alpha) can be expressed as rational functions in \alpha with coefficients in K, specifically of the form p(\alpha)/q(\alpha), where p(x) and q(x) are polynomials in K and q(\alpha) \neq 0.^[17] If \alpha is algebraic over K, then the elements are precisely the linear combinations \sum_{i=0}^{n-1} c_i \alpha^i with c_i \in K, where n is the degree of the minimal polynomial of \alpha over K.^[18] The primitive element theorem states that every finite separable extension L/K is simple, i.e., L = K(\alpha) for some \alpha \in L.^[16] The proof relies on the linear independence of certain conjugates of the generators, allowing the construction of a primitive element as a suitable linear combination that avoids finitely many "bad" values which would cause dependencies.^[18] In particular, all finite extensions in characteristic zero are simple, as separability holds automatically there.^[16] Not all finite extensions are simple; for instance, the extension \mathbb{F}_p(X,Y)/\mathbb{F}_p(X^p, Y^p) has degree p^2 but requires at least two generators and admits no primitive element.^[17] Infinite extensions can also be simple, such as the transcendental extension \mathbb{Q}(\pi)/\mathbb{Q}.^[16] For a simple finite extension L = K(\alpha), the degree [L:K] equals the degree of the minimal polynomial of \alpha over K.^[18] This follows from the fact that \{1, \alpha, \dots, \alpha^{n-1}\} forms a basis for L as a vector space over K, where n = [L:K].^[17]

Tower law for degrees

The tower law, also known as the multiplicativity of degrees, states that for a tower of field extensions K \subseteq M \subseteq L, the degree of the overall extension satisfies [L : K] = [L : M] \cdot [M : K], provided the individual degrees are finite.^[19] This relation extends multiplicatively to any finite tower of extensions by induction on the length of the chain.^[19] To sketch the proof for the finite case, suppose \{u_i \mid 1 \leq i \leq [M : K]\} is a basis for M as a vector space over K, and \{v_j \mid 1 \leq j \leq [L : M]\} is a basis for L as a vector space over M. Then the set \{u_i v_j\} forms a basis for L as a vector space over K, establishing the dimension equality [L : K] = [L : M] \cdot [M : K].^[19] This multiplicativity arises specifically because field extensions behave as vector spaces, where dimensions multiply in towers; it does not hold in general for extensions of rings, where modules may lack unique ranks.^[20] In the infinite case, if at least one of [L : M] or [M : K] is infinite, then [L : K] is also infinite, and the equality holds in the sense of cardinal multiplication of the dimensions as vector spaces.^[20] The tower law facilitates degree computations in composite extensions; for instance, adjoining square roots successively yields [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2 and [\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}(\sqrt{2})] = 2, so by the tower law, [\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 4.^[21]

Finite versus infinite extensions

A field extension L/K is finite if its degree [L : K] is a finite positive integer, meaning L is a finite-dimensional vector space over the base field K. Every finite extension is algebraic, in the sense that every element of L satisfies a polynomial equation with coefficients in K.^[22] This algebraicity follows from the fact that a basis of L over K allows any element to be expressed as a linear combination, leading to a characteristic polynomial of bounded degree.^[22] An extension L/K is infinite if [L : K] = \infty, so L is not finite-dimensional over K. Infinite extensions may be algebraic, as in the case of the algebraic numbers \overline{\mathbb{Q}} over the rationals \mathbb{Q}, or transcendental, involving elements not satisfying any polynomial over K, such as \mathbb{R}/\mathbb{Q} which contains elements like \pi.^[22] Every field extension L/K possesses a transcendence basis, defined as a maximal subset of L that is algebraically independent over K; the cardinality of any such basis is the transcendence degree of the extension. The transcendence degree is zero if and only if the extension is algebraic, and the extension is finite if and only if the transcendence degree is zero and the resulting algebraic extension has finite degree.^[23] Infinite extensions thus allow for transcendence bases of positive cardinality, enabling the decomposition of L into a transcendental part followed by an algebraic extension. Finite extensions exhibit properties analogous to those of finite-dimensional vector spaces, such as the existence of bases, traces, norms, and determinants for K-linear maps from L to itself.^[22] These features do not hold for infinite extensions, where linear maps may lack such invariants. For example, the tower law states that degrees multiply in finite towers of extensions, but infinite degrees prevent similar multiplicative behavior.^[22] In certain contexts, such as number fields, finite extensions L/\mathbb{Q} ensure that the integral closure of \mathbb{Z} in L is finitely generated as a \mathbb{Z}-module.^[24]

Illustrative Examples

Rational to algebraic number fields

A fundamental example of a finite algebraic extension is \mathbb{Q}(\sqrt{2})/\mathbb{Q}, which has degree 2.^[19] The standard basis for this extension as a vector space over \mathbb{Q} is \{1, \sqrt{2}\}, and the minimal polynomial of \sqrt{2} over \mathbb{Q} is x^2 - 2.^[19] Every element in \mathbb{Q}(\sqrt{2}) can be uniquely expressed as a + b\sqrt{2} with a, b \in \mathbb{Q}. For extensions adjoining multiple square roots, consider \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}. This has degree 4, obtained via the tower \mathbb{Q} \subseteq \mathbb{Q}(\sqrt{2}) \subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3}), where each step has degree 2.^[25] The extension is simple, generated by the primitive element \sqrt{2} + \sqrt{3}, whose minimal polynomial over \mathbb{Q} is x^4 - 10x^2 + 1.^[25] More generally, quadratic fields take the form \mathbb{Q}(\sqrt{d})/\mathbb{Q} for square-free integers d \neq 1, each of degree 2 over \mathbb{Q}.^[26] The discriminant of such a field is \Delta = d if d \equiv 1 \pmod{4} and \Delta = 4d otherwise, which measures the "ramification" in the extension.^[27] The ring of integers \mathcal{O}_K is \mathbb{Z}[\sqrt{d}] if d \equiv 2, 3 \pmod{4} and \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right] if d \equiv 1 \pmod{4}.^[27] All quadratic extensions are simple, as guaranteed by the primitive element theorem for finite separable extensions.^[18] Quadratic fields play a key role in number theory, particularly for solving Diophantine equations such as Pell's equation x^2 - d y^2 = \pm 1, where solutions correspond to units in the ring of integers.^[28] In quadratic fields, the norm map N_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}(a + b\sqrt{d}) = a^2 - d b^2 and trace map \mathrm{Tr}_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}(a + b\sqrt{d}) = 2a provide multiplicative and additive invariants, previewing their generalization via field automorphisms in broader contexts.^[29]

Function fields and transcendental extensions

A classic example of a transcendental field extension is the rational function field \mathbb{C}(t) over the complex numbers \mathbb{C}, where t is an indeterminate, consisting of all quotients of polynomials in t with coefficients in \mathbb{C}.^[30] This extension has transcendence degree 1, meaning \{t\} forms a transcendence basis, and it is of infinite degree as a vector space over \mathbb{C}.^[30] Unlike algebraic extensions, t satisfies no minimal polynomial over \mathbb{C}, highlighting its transcendental nature.^[31] Elements of \mathbb{C}(t) are formal expressions f(t)/g(t), where f, g \in \mathbb{C} and g \neq 0, with equality defined by cross-multiplication after clearing common factors.^[32] The extension admits no finite basis as a vector space over \mathbb{C}, as powers of t and their inverses generate infinitely many linearly independent elements.^[33] A real analog is \mathbb{R}(x)/\mathbb{R}, the field of rational functions in x over the reals, which similarly exhibits transcendence degree 1 and models real rational functions on the line.^[32] Function fields like \mathbb{C}(t)/\mathbb{C} play a central role in algebraic geometry, where they model the rational functions on algebraic curves, with the transcendence degree corresponding to the dimension of the curve (here, 1 for a curve).^[34] In complex analysis, such fields relate to meromorphic functions on Riemann surfaces, providing a field-theoretic framework for studying holomorphic forms and divisors on these surfaces.^[35] Transcendental extensions of this type are inherently non-algebraic, distinguishing them from finite extensions where every element satisfies a polynomial equation over the base field.^[30]

Cyclotomic extensions

A cyclotomic extension is the field extension \mathbb{Q}(\zeta_n)/\mathbb{Q}, obtained by adjoining a primitive nth root of unity \zeta_n to the rational numbers \mathbb{Q}, where \zeta_n satisfies \zeta_n^n = 1 and no smaller positive exponent works.^[36] The degree of this extension is [\mathbb{Q}(\zeta_n) : \mathbb{Q}] = \phi(n), where \phi denotes Euler's totient function, which counts the number of integers up to n that are coprime to n.^[36] The minimal polynomial of \zeta_n over \mathbb{Q} is the monic nth cyclotomic polynomial \Phi_n(x), defined as \Phi_n(x) = \prod (x - \zeta), where the product runs over all primitive nth roots of unity \zeta, and it has degree \phi(n).^[37] For example, when n=5, \phi(5) = 4, so [\mathbb{Q}(\zeta_5) : \mathbb{Q}] = 4, and the minimal polynomial is \Phi_5(x) = x^4 + x^3 + x^2 + x + 1.^[37] The extension \mathbb{Q}(\zeta_n)/\mathbb{Q} is Galois, hence normal and separable, with Galois group \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) isomorphic to the multiplicative group of units modulo n, denoted (\mathbb{Z}/n\mathbb{Z})^\times, which is abelian.^[36] This abelian structure makes cyclotomic extensions fundamental in class field theory and Galois theory. Cyclotomic fields have been instrumental in proofs of Fermat's Last Theorem, especially in Kummer's approach using unique factorization failures in rings of cyclotomic integers and subsequent developments involving class numbers and units.^[38] The field \mathbb{Q}(\zeta_n) contains all mth roots of unity for m dividing n, as these are polynomials in \zeta_n.^[36] For a fixed prime p, the cyclotomic fields \mathbb{Q}(\zeta_{p^k}) for k = 1, 2, \dots form an infinite tower of extensions, each of degree p over the previous one, leading to the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q}.^[38] An explicit power basis for \mathbb{Q}(\zeta_n) as a vector space over \mathbb{Q} is \{1, \zeta_n, \zeta_n^2, \dots, \zeta_n^{\phi(n)-1}\}, reflecting the simplicity of the extension generated by a single algebraic element.^[36]

Algebraic and Transcendental Extensions

Algebraic extensions

An element \alpha in an extension field L of a field K is said to be algebraic over K if there exists a non-zero polynomial f(x) \in K such that f(\alpha) = 0.^[3] The minimal polynomial of \alpha over K, denoted m_\alpha(x), is the monic irreducible polynomial in K of least degree that has \alpha as a root.^[39] Thus, m_\alpha(\alpha) = 0, and every algebraic element satisfies its minimal polynomial.^[40] A field extension L/K is algebraic if every element of L is algebraic over K.^[20] Such extensions include simple extensions obtained by adjoining a single algebraic element \alpha, where the degree [K(\alpha):K] equals the degree of the minimal polynomial \deg(m_\alpha).^[3] An algebraic extension L/K is finite if and only if [L:K] is finite, in which case L is generated by finitely many algebraic elements over K.^[41] The algebraic extensions of K are closed under finite unions, meaning the compositum of finitely many algebraic extensions of K is again algebraic over K.^[20] The algebraic closure of K, denoted \bar{K}, is a maximal algebraic extension of K that is algebraically closed, meaning every non-constant polynomial in \bar{K} has a root in \bar{K}.^[41] Any two algebraic closures of K are isomorphic over K.^[41] For example, the algebraic closure \bar{\mathbb{Q}} of the rationals \mathbb{Q} consists precisely of the algebraic numbers, which form an algebraically closed field of infinite degree over \mathbb{Q}.^[40] In algebraic extensions, particularly over \mathbb{Q}, an element \alpha is integral over \mathbb{Z} (an algebraic integer) if its minimal polynomial over \mathbb{Q} has integer coefficients and is monic.^[39] For a basis of a finite algebraic extension, the discriminant measures the "overlap" between basis elements and is defined as the determinant of the trace form matrix on the basis.^[20]

Transcendental extensions

In field theory, an element \alpha in a field extension L/K is called transcendental over K if it is not algebraic over K, meaning there does not exist a nonzero polynomial f(x) \in K such that f(\alpha) = 0.^[42] A subset S \subseteq L is algebraically independent over K if no finite nonempty subset of S satisfies a nontrivial polynomial relation with coefficients in K, or equivalently, the evaluation map K[x_s \mid s \in S] \to L sending each x_s to s is injective.^[42] A transcendence basis for L/K is a maximal algebraically independent subset B \subseteq L over K, and every such basis has the same cardinality, called the transcendence degree of L/K, denoted \operatorname{tr.deg}(L/K).^[43] A key property of field extensions is that every extension L/K admits a transcendence basis B, and moreover, L is algebraic over the purely transcendental extension K(B), where K(B) is the field generated by K and the elements of B.^[42] Thus, any field extension can be viewed as a tower consisting of a purely transcendental extension followed by an algebraic extension.^[43] For example, if u is transcendental over \mathbb{C}(t), then \operatorname{tr.deg}(\mathbb{C}(t,u)/\mathbb{C}) = 2, with \{t, u\} serving as a transcendence basis.^[43] Purely transcendental extensions of finite transcendence degree n are isomorphic to the rational function field K(x_1, \dots, x_n).^[42] The transcendence degree can be infinite, as in the case of \mathbb{R}/\mathbb{Q}, where \operatorname{tr.deg}(\mathbb{R}/\mathbb{Q}) equals the cardinality of the continuum.^[44] In general, if \operatorname{tr.deg}(L/K) > 0, then the extension degree [L : K] is infinite, since even a purely transcendental extension of positive finite degree has infinite degree over K.^[42] Transcendence degrees play a crucial role in the study of function fields; for instance, Lüroth's theorem states that any subfield L with K \subsetneq L \subsetneq K(t) (where t is transcendental over K) has transcendence degree 1 over K and is itself a simple transcendental extension K(u) for some u \in L.^[45] This highlights the uniqueness of transcendence degree 1 extensions in the univariate case, with applications to the birational geometry of curves.^[45]

Special Types of Extensions

Normal extensions

A field extension L/K is called normal if it is algebraic and every irreducible polynomial f \in K that has at least one root in L splits completely into linear factors in L.^[46] This condition ensures that L contains all conjugates of any element over K, preserving the full structure of minimal polynomials.^[47] For finite extensions, L/K is normal if and only if L is the splitting field over K of some polynomial f \in K.^[46] In the general algebraic case, including infinite extensions, L/K is normal if and only if L is the splitting field over K of some family of polynomials in K.^[48] Normal extensions exhibit several key properties. The intersection of any finite collection of normal extensions of K contained in a common algebraic closure is again a normal extension of K.^[49] Moreover, if L/K is normal, then the Galois group \mathrm{Gal}(L/K) (defined as the group of K-automorphisms of L) acts transitively on the roots in L of any irreducible polynomial in K.^[50] Finite normal extensions are Galois precisely when they are also separable.^[51] For example, cyclotomic extensions \mathbb{Q}(\zeta_n)/\mathbb{Q}, where \zeta_n is a primitive nth root of unity, are normal because they are the splitting fields of the irreducible cyclotomic polynomials \Phi_n(x) \in \mathbb{Q}.^[47] In contrast, the extension

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

, adjoining only the real cube root of 2, is not normal, as the minimal polynomial x^3 - 2 has one root in the extension but its other two complex roots are missing.^[46]

Separable extensions

A polynomial f \in K[X] over a field K is called separable if it has distinct roots in a splitting field over K, meaning no root has multiplicity greater than one; equivalently, f and its formal derivative f' are coprime in K[X], so \gcd(f, f') = 1.^[52]^[53] An algebraic extension L/K is separable if every element \alpha \in L has a separable minimal polynomial over K; for finite extensions, this holds if and only if the number of distinct K-embeddings of L into an algebraic closure of K equals the degree [L:K].^[52]^[54] In fields of characteristic zero, every irreducible polynomial is separable (since f' \neq 0), so every algebraic extension is separable.^[52]^[53] In characteristic p > 0, inseparable extensions exist; a basic example is the purely inseparable extension \mathbb{F}_p(t)/\mathbb{F}_p(t^p) of degree p, where t is transcendental over \mathbb{F}_p and the minimal polynomial of t over \mathbb{F}_p(t^p) is X^p - t^p = (X - t)^p, which has a multiple root.^[55]^[52] The separable closure of a field K exists as a Galois extension inside an algebraic closure of K, consisting of all elements separable over K; it is the compositum of all finite separable extensions of K.^[56] Moreover, the compositum of any two separable extensions of K contained in a common extension is itself separable over K.^[52]^[57] Separable polynomials split into distinct linear factors in normal extensions, distinguishing separability from normality by focusing on root multiplicity rather than completeness of splitting.^[52]

Galois extensions

A Galois extension is a finite field extension L/K that is both normal and separable, meaning every irreducible polynomial over K with a root in L splits completely into linear factors in L, and the minimal polynomial of every element of L over K has distinct roots.^[58] Equivalently, L is the splitting field over K of a separable polynomial.^[59] The associated Galois group \mathrm{Gal}(L/K) is defined as the group \mathrm{Aut}_K(L) of all field automorphisms of L that fix K pointwise, and for a finite Galois extension, the order of this group equals the degree of the extension: |\mathrm{Gal}(L/K)| = [L:K].^[60]^[58] The fundamental theorem of Galois theory establishes a bijective, order-reversing correspondence between the subfields of L containing K and the subgroups of \mathrm{Gal}(L/K).^[58] Specifically, each subgroup H \leq \mathrm{Gal}(L/K) corresponds to its fixed field L^H = \{ x \in L \mid \sigma(x) = x \ \forall \sigma \in H \}, and each intermediate field K \subseteq M \subseteq L corresponds to the subgroup \mathrm{Gal}(L/M); the degree [L:M] equals |H| where H = \mathrm{Gal}(L/M), and normal subgroups correspond to Galois subextensions.^[61]^[58] This lattice isomorphism provides a deep connection between the algebraic structure of the field extension and the group-theoretic structure of its automorphism group.^[58] A field extension L/K is Galois if and only if it is finite, normal, and separable.^[58] For infinite Galois extensions, the theory extends by equipping the Galois group with the Krull topology, making it a profinite group (an inverse limit of finite groups), and the fundamental theorem holds for closed subgroups and their fixed fields.^[62]^[63] In such cases, the extension is the union of finite Galois subextensions, and the profinite structure captures the topology induced by open normal subgroups corresponding to finite quotients.^[58] Key properties of Galois extensions include the fact that the discriminant of a separable polynomial defining the extension (or more generally, of the extension itself via the different ideal) is nonzero, reflecting the separability condition.^[58] Galois theory plays a central role in determining solvability by radicals: a polynomial over a field of characteristic zero is solvable by radicals if and only if the Galois group of its splitting field is a solvable group.^[58] For example, the extension \mathbb{Q}(\sqrt{2})/\mathbb{Q} is Galois with \mathrm{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}, generated by the automorphism sending \sqrt{2} to -\sqrt{2}.^[59]^[58]

Generalizations and Applications

Ring extensions

A ring extension consists of commutative rings A and B with a ring homomorphism \iota: A \to B such that B is an A-algebra via \iota, meaning B is also a module over A through the action induced by \iota.^[64] In this setup, B need not be a field even if A is, and elements of B are not necessarily invertible, unlike in field extensions where the codomain inherits the division ring structure.^[65] An integral extension is a special case where every element b \in B is integral over A, satisfying a monic polynomial f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in A with f(b) = 0.^[64] Key results include the going-up theorem, which states that for a prime ideal \mathfrak{p} \subset A, there exists a prime ideal \mathfrak{q} \subset B such that \mathfrak{q} \cap A = \mathfrak{p}, and chains of primes in A can be lifted to chains in B of the same length; the going-down theorem holds under additional conditions like normality of A.^[66] These theorems highlight how integral extensions preserve certain ideal-theoretic properties, such as lying-over (every prime in A contracts from some prime in B) and incomparability (corresponding primes have the same height).^[67] In the context of Dedekind domains, if A is a Dedekind domain and B is its integral closure in a finite extension of the fraction field of A, then B is also a Dedekind domain, preserving unique factorization of ideals into primes.^[24] This is crucial in number theory, where for extensions of number fields, ramification occurs when a prime ideal \mathfrak{p} \subset A factors in B with some prime powers having exponent greater than 1, measured by the ramification index e(\mathfrak{q}/\mathfrak{p}) for \mathfrak{q} \cap A = \mathfrak{p}.^[68] Notably, field extensions are integral precisely when they are algebraic, as every algebraic element satisfies a monic minimal polynomial over the base field.^[64] Unlike field extensions, where B is automatically a finite-dimensional vector space over A if algebraic, ring extensions generally lack such a structure; instead, one considers whether B is flat or projective as an A-module, properties that ensure exactness of tensor products or lifting of projectives, but these do not hold universally without additional assumptions like freeness.^[69]

Extension of scalars

In the context of a field extension L/K, the extension of scalars provides a method to change the base field of a vector space while preserving its module structure. Given a vector space V over K, the extended space V_L = V \otimes_K L is naturally an L-vector space, where the L-action is defined by \ell \cdot (v \otimes \ell') = v \otimes (\ell \ell') for v \in V, \ell, \ell' \in L.^[70]^[71] This construction arises from viewing L as a K-module via the inclusion and forming the tensor product, which endows V_L with the structure of an L-module.^[70] Key properties of this extension include the preservation of dimension for finite-dimensional spaces and conditions related to flatness. If \dim_K V = n < \infty, then \dim_L V_L = n, with a basis for V_L given by \{1 \otimes b_i \mid b_i\} where \{b_i\} is a basis for V.^[70] Moreover, since any field extension L/K renders L flat as a K-module (as it is a free K-module of rank equal to the degree if finite, or more generally a vector space), the functor V \mapsto V_L is exact, preserving exact sequences of K-vector spaces.^[72]^[71] This mechanism is foundational in algebraic geometry, where it facilitates the study of schemes over varying base fields by pulling back quasicoherent sheaves or vector bundles along base change morphisms, and plays a central role in descent theory for ensuring properties descend effectively under faithfully flat extensions.^[71]^[73] For instance, consider a \mathbb{Q}-vector space V; extending scalars to \mathbb{R} yields V_{\mathbb{R}} = V \otimes_{\mathbb{Q}} \mathbb{R}, which is an \mathbb{R}-vector space of the same dimension as V, effectively "complexifying" or realifying the structure while maintaining linear independence relations.^[70] Field extensions naturally induce scalar extensions on associated modules, transforming K-linear structures into L-linear ones compatibly with the extension map.^[71]

Applications in algebra and geometry

Field extensions play a crucial role in algebra, particularly in determining the solvability of polynomial equations by radicals, as established through Galois theory. In this framework, a polynomial is solvable by radicals if and only if its splitting field over the rationals has a solvable Galois group, a result originating from Galois's work on the quintic equation. This criterion not only resolves classical problems like the unsolvability of the general quintic but also extends to broader classes of equations, enabling the classification of solvable extensions via group-theoretic properties. In fields of characteristic p, Artin-Schreier theory provides an analogous tool for constructing cyclic extensions of degree p, where the extensions are generated by roots of equations of the form x^p - x = a for a in the base field not already in the image of the Artin-Schreier map. This construction is fundamental for understanding the structure of extensions in positive characteristic, paralleling Kummer theory in characteristic zero and facilitating the study of differential equations over such fields.^[74] In number theory, field extensions underpin class field theory, which describes all abelian extensions of a number field as corresponding to ideals in its ring of integers via the Artin map. This generalization of Galois theory to infinite abelian groups connects algebraic structures to arithmetic, with applications to the distribution of primes in extensions. L-functions, such as Dirichlet L-functions for cyclotomic extensions, encode information about the arithmetic of these fields, including their zeta functions and regulator constants, which are central to the study of units and class numbers.^[75] Geometric applications arise in the study of algebraic curves, where the function field of a curve over a base field captures its birational geometry, with extensions corresponding to branched covers. The Riemann-Hurwitz formula quantifies the ramification in such covers, relating the genus of the extension field to that of the base via the degree and ramification indices: for a separable extension K/k of function fields of curves, 2g_K - 2 = [K:k](2g_k - 2) + \sum (e_P - 1), where g denotes genus and e_P the ramification index at place P.^[76] Modern developments highlight the role of p-adic extensions in local number theory, where completions of global fields yield insights into global arithmetic via local-global principles. Anabelian geometry, as pursued in Grothendieck's program, posits that the algebraic fundamental group of a variety, derived from étale extensions, recovers the variety's isomorphism type, bridging field extensions to topological reconstruction. Étale cohomology further employs Galois extensions to define sheaf cohomology on algebraic stacks, providing tools for motivic cohomology and arithmetic geometry beyond classical topology.^[77] Hilbert's 13th problem concerns the decomposability of algebraic functions into superpositions of functions of fewer variables, with resolutions involving towers of algebraic extensions showing that certain polynomials require towers of height greater than 1, as measured by the resolvent degree.^[78] The Ax-Lindemann-Weierstrass theorem implies that if α is algebraic over the rationals, then e^{iα} is transcendental over ℚ(π) unless α/π is rational, linking algebraic and transcendental extensions in complex analysis.

References

[1]
[PDF] RES.18-012 (Spring 2022) Lecture 25: Field Extensions
Definition 25.12. For a polynomial P ∈ F[x], a splitting field of P is an extension E/F such that: 1. P splits as a product of linear factors in E[x];. 2. E = F ...
[2]
[PDF] Abstract Algebra. Math 6320. Bertram/Utah 2022-23. Fields We have ...
Definition. (a) An inclusion K ⊂ L of fields is a field extension, in which case L is an algebra and a vector space over K, and if the dimension of L is finite ...
[3]
[PDF] an introduction to the theory of field extensions - UChicago Math
This document introduces the theory of rings, then elaborates on the theory of field extensions, and examines consequences of the subject.
[4]
[PDF] Notes on graduate algebra - Department of Mathematics
These graduate algebra notes cover fields and Galois theory, including field extensions, algebraic closure, splitting fields, and separable polynomials.<|control11|><|separator|>
[5]
Extension Field -- from Wolfram MathWorld
An extension field K/F occurs when F is a subfield of K. For example, complex numbers are an extension of real numbers.
[6]
[PDF] Part VI. Extension Fields
Feb 16, 2013 · Definition 29.6. An element α of an extension field E of a field F is algebraic over F if f(α) = 0 for some nonzero f( ...
[7]
Section 9.2 (09FC): Basic definitions—The Stacks project
1. A field is a nonzero ring where every nonzero element is invertible. Given a field a subfield is a subring that is itself a field.
[8]
[PDF] reu apprentice class #9
Jul 8, 2011 · A subset F ⊆ H of a field H is a subfield if it is a field under the operations of H, i.e., if F is closed under sums, products, negation, ...
[9]
[PDF] Math 430 – Problem Set 7 Solutions
Apr 22, 2016 · Suppose that F is a field, and let E be the intersection of all subfields of F. We show that E is the unique prime subfield of F. Note that 0, 1 ...
[10]
Definition 9.5.1 (09FR)—The Stacks project
The prime subfield of F is the smallest subfield of F which is either \mathbf{Q} \subset F if the characteristic is zero, or \mathbf{F}_ p \subset F if the ...
[11]
[PDF] Section V.1. Field Extensions
Dec 30, 2023 · A field F is an extension field of field K if K is a subfield of F. Algebraic extensions have elements algebraic over K, while transcendental ...
[12]
[PDF] Lecture 5 - Math 5111 (Algebra 1)
Definition If F is a field, we say a subset S of F is a subfield if S is itself a field under the same operations as F.
[13]
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra:_Theory_and_Applications_(Judson)/21:_Fields/21.01:_Extension_Fields
[14]
[PDF] Abstract Algebra, Lecture 14 - Field extensions - Linköpings universitet
The dimension is denoted by [F : E], and refered to as the degree of the extension. If this dimension is finite, then the extension is said to be finite.
[15]
[PDF] 1. the degree of a field extension - Galois theory lecture summary
Mar 5, 2018 · This extension presents a greater challenge than the previous one. The key observation is that while Q is countably infinite, R is uncountable.
[16]
Transcendence Degree -- from Wolfram MathWorld
The transcendence degree of Q(pi), sometimes called the transcendental degree, is one because it is generated by one extra element.
[17]
[PDF] Mathematics 6310 The Primitive Element Theorem Ken Brown ...
Given a field extension K/F, an element α ∈ K is said to be separable over F if it is algebraic over F and its minimal polynomial over F is separable.
[18]
[PDF] Lecture 21 - Math 5111 (Algebra 1)
In general, an element α generating the extension K/F is called a primitive element for K/F, whence the name “primitive element theorem”.
[19]
[PDF] The Primitive Element Theorem.
The Primitive Element Theorem. Assume that F and K are subfields of C and that K/F is a finite extension. Then K = F(θ) for some element θ in K.
[20]
[PDF] 29 Extension Fields - UCI Mathematics
This is an example of a simple extension, where we adjoin a single element to a given field and use the field operations to produce as many new elements as ...
[21]
[PDF] Contents 2 Fields and Field Extensions - Evan Dummit
◦ Proof: If F(α) is algebraic then α ∈ F(α) is algebraic over F. If the minimal polynomial for α has degree n, then as we showed earlier, [F(α) : F] = n, so ...
[22]
[PDF] ALGEBRA HW 6 547.4 Prove that Q( √ 2) and Q( √ 3) ar not ...
547.4. Prove that Q(. √. 2) and Q(. √. 3) ar not isomorphic. Proof. Suppose there exists an isomorphism φ : Q(. √. 2) → Q(. √. 3) ...<|control11|><|separator|>
[23]
[PDF] Lecture 6 - Math 5111 (Algebra 1)
Definition. The field extension K/F is algebraic if every α ∈ K is algebraic over F: in other words, if every α is a root of a nonzero polynomial in F[x]. Our ...Missing: abstract | Show results with:abstract
[24]
[PDF] Lecture 10 - Math 5111 (Algebra 1)
Let K/F be a field extension. A transcendence base for K/F is an algebraically independent subset S of K that is maximal in the set of all algebraically ...
[25]
[PDF] 5 Dedekind extensions - MIT Mathematics
Sep 22, 2016 · In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also a Dedekind ...Missing: abstract | Show results with:abstract
[26]
[PDF] Section 6: Field and Galois theory
The set of all automorphisms of a field forms a group under composition. Definition ... K ∩ R is a subfield of R if and only if K is a subfield of C. 2 ...<|control11|><|separator|>
[27]
[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.
[28]
[PDF] Explicit Methods in Algebraic Number Theory
1.8 Ring of Integers of Quadratic Number Fields. Let us consider a quadratic number field K = Q(. √ d) with d square free. Let α = a+b. √ d ∈ OK, then its ...
[29]
https://kconrad.math.uconn.edu/blurbs/galoistheory/tracenorm.pdf
[30]
[PDF] TRACE AND NORM 1. Introduction Let L/K be a finite extension of ...
Introduction. Let L/K be a finite extension of fields, with n = [L : K]. We will associate to this extension two important functions L → K, called the trace ...
[31]
Section 9.26 (030D): Transcendence—The Stacks project
Let K/k be a field extension. The transcendence degree of K over k is the cardinality of a transcendence basis of K over k. It is denoted \text{trdeg}_ k(K).
[32]
[PDF] Transcendental extensions - Brandeis
Purely transcendental extensions. These are field extensions of k which are isomorphic to a fraction field of a polynomial ring:.
[33]
Function fields - Purdue Math
In particular, it is a field of transcendence degree 1 over C, which means that it is an algebraic extension of a field of rational functions in one variable.
[34]
[PDF] Contents - UChicago Math
Now we prove that the transcendence degree of a field extension is independent of choice of basis. Theorem 5.5 Let F/E be a field extension. Any two ...
[35]
Section 53.2 (0BXX): Curves and function fields—The Stacks project
A variety is a curve if and only if its function field has transcendence degree 1, see for example Varieties, Lemma 33.20.3.
[36]
[PDF] A concise course in complex analysis and Riemann surfaces ...
It is clear that the meromorphic functions on a Riemann surface form a field. One refers to this field as the function field1 of a surface M. 2. Examples. In ...
[37]
[PDF] cyclotomic extensions - keith conrad
The term cyclotomic means “circle-dividing,” which comes from the fact that the nth roots of unity in C divide a circle into n arcs of equal length, as in ...
[38]
Cyclotomic Polynomial -- from Wolfram MathWorld
Cyclotomic Polynomial ; Phi_5(x), = x^4+x^3+x^2+x+1. (9) ; Phi_6(x), = x^2-x+1. (10) ; Phi_7(x), = x^6+x^5+x^4+x^3+. (11).
[39]
Introduction to Cyclotomic Fields - SpringerLink
In stock Free deliveryStarting at an elementary level, the volume covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p- ...
[40]
[PDF] FIELD THEORY 1. Fields, Algebraic and Transcendental Elements ...
If F is a field, we know that the intersection F0 of all subfields of F is again a field—called the prime subfield of F. It is the smallest subfield of F. The ...Missing: definition | Show results with:definition
[41]
[PDF] Section VI.31. Algebraic Extensions
Mar 21, 2015 · An extension field E of field F is an algebraic extension of F if every element in E is algebraic over F. Example. Q(√2) and Q(√3) are ...
[42]
[PDF] 12.1 Field extensions
Oct 17, 2013 · For every field k there exists an extension ¯k/k with ¯k algebraically closed; such a ¯k is called an algebraic closure of k, and all such ¯k ...
[43]
[PDF] transcendence degree - Anand Deopurkar
Theorem. Every field extension E/F is a purely transcendental ex- tension followed by an algebraic extension. Proof. Take a transcendence base S for E/F.
[44]
[PDF] 18.782 Arithmetic Geometry Lecture Note 12 - MIT OpenCourseWare
Oct 17, 2013 · Definition 12.3. The transcendence degree of a field extension L/K is the cardinality of any (hence every) transcendence basis for L/k.
[45]
[PDF] Zorn's lemma and some applications, II - Keith Conrad
The extensions C/Q and R/Q have transcendence degree equal the car- dinality of the continuum. 4. More extension problems using Zorn's lemma. Zorn's lemma is ...
[46]
[PDF] Lüroth's Theorem, and some related results
A simple transcendental extension of a field k means an extension of the form k(t) where t is transcendental over k; in other words, an extension isomorphic to ...<|control11|><|separator|>
[47]
[PDF] 9. Normal and Separable extensions - UCSD Math
Normal and Separable extensions. Now we turn to the question, given a field extension, when is there some polynomial for which it is a splitting field?
[48]
[PDF] splitting fields and normal extensions - Galois theory lecture summary
Apr 30, 2018 · CHARACTERIZING NORMAL EXTENSIONS. Recall that L/K is a normal extension if and only if every irreducible f ∈ K[x] either has no roots in.
[49]
[PDF] Field Theory, Part 2: Splitting Fields; Algebraic Closure - Jay Havaldar
Definition: If K is an algebraic extension of F which is a splitting field over F for a collection of polynomials in F[x], then K is called a normal extension ...
[50]
Field Theory [2 ed.] 0387276777, 9780387276779 - DOKUMEN.PUB
... intersection of normal extensions is normal, so 5 ~ ¸2 “ , 2 - and - x 2¹ Definition Let - , - . The normal closure of , over - in - is the smallest ...
[51]
3.3 Splitting fields and normal extensions - Fiveable
Galois group of normal extension K/F acts transitively on roots of any irreducible polynomial in F[x] with one root in K; Normal extensions remain closed ...
[52]
[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 ...
[53]
[PDF] Mathematics 3360 Separable polynomials Ken Brown, Cornell ...
f is separable if and only if f and its derivative f0 are coprime in. F[x]. So we have a simple test (via the Euclidean algorithm), that doesn't require ...
[54]
[PDF] Separability of finite field extensions - Brown Math Department
If K ⊃ k is a finite-degree extension of fields, then the ex- tension is separable if the separable degree is equal to the field extension degree.
[55]
[PDF] Purely inseparable field extensions - ETH Zürich
Fp(t)/Fp(tp) is a purely inseparable field extension, where t is tran- scendental over Fp. Proof. Let x ∈ Fp(t), then x = X. 0≤i≤p− ...
[56]
[PDF] 4 Étale algebras, norm and trace - 4.1 Separability - MIT Mathematics
Sep 20, 2021 · Definition 4.18. A field K is perfect if every algebraic extension of K is separable. All fields of characteristic zero are perfect. Perfect ...<|control11|><|separator|>
[57]
[PDF] Lecture 9 - Math 5111 (Algebra 1)
Definition If K/F is an algebraic extension, then α ∈ K is separable over F if α is algebraic over K and its minimal polynomial m(x) over F is a separable ...
[58]
[PDF] Fields and Galois Theory - James Milne
Galois extension of 𝐹 (possibly infinite) with Galois group 𝐺. ... According to PARI this has no nonzero rational points, and so the discriminant can't be.
[59]
Galois Extension Field -- from Wolfram MathWorld
1. K is the splitting field for a collection of separable polynomials. When K is a finite extension, then only one separable polynomial is necessary.
[60]
Galois Group -- from Wolfram MathWorld
### Definition of Galois Group
[61]
Fundamental Theorem of Galois Theory -- from Wolfram MathWorld
For a Galois extension field of a field , the fundamental theorem of Galois theory states that the subgroups of the Galois group correspond with the subfields ...
[62]
Section 9.22 (0BMI): Infinite Galois theory—The Stacks project
Let L/K be a Galois extension. Let G = \text{Gal}(L/K) be the Galois group viewed as a profinite topological group (Lemma 9.22.1). Then we have K = L^ G and ...
[63]
[PDF] Infinite Galois theory, Stone spaces, and profinite groups
The Galois group of an infinite-dimensional normal algebraic extension is easily described in terms of the Galois groups of its subextensions of finite degree.
[64]
Ideal Theory of Commutative Rings - Northern Illinois University
An element u T is said to be integral over R if there exists a monic polynomial f(x) R[x] such that f(u)=0. The ring T is said to be an integral extension of R ...<|control11|><|separator|>
[65]
10.36 Finite and integral ring extensions - Stacks Project
Integral closure commutes with localization: If A \to B ... If R \subset K and K is finite over R, then R is a field and K is a finite algebraic extension.
[66]
[PDF] The Going-Up Theorem
Throughout this discussion, we fix an integral ring extension A ⊂ B. Theorem 1 (Going Up) Suppose P ⊂ A is a prime ideal. Then there exists a prime ideal Q ⊂ B.
[67]
[PDF] 9. Integral Ring Extensions
There are four main geometric results on integral ring extensions in the above spirit; they are com- monly named Lying Over, Incomparability, Going Up, and ...
[68]
[PDF] Ramification in algebraic number theory and dynamics
Jan 24, 2019 · Let L/K be an extension of number fields, A ⊂ K a Dedekind domain with field of fractions K and B its integral closure in L. If q is a prime in.
[69]
[PDF] More on Flatness - Stacks Project
Introduction. 057N In this chapter, we discuss some advanced results on flat modules and flat mor- phisms of schemes and applications.
[70]
[PDF] 1 Extension of Scalars
Remark. Extension of scalars is “functorial”. That is, given an R-linear map f : M −! N we have the induced map id⊗f : S ⊗R M −!
[71]
extension of scalars in nLab
### Summary of Extension of Scalars (nLab)
[72]
[PDF] EXTENSION AND RESTRICTION OF SCALARS Let f
−−→ f∗(VC)). 1 Here we use that f : R → C is flat, as is any field extension. Indeed, extension of scalars along a ring homomorphism f : A → B preserves ...
[73]
10.39 Flat modules and flat ring maps - Stacks Project
An R-module M is flat if N \mapsto N \otimes _ RM is also left exact, ie, if it is exact. Here is the precise definition.