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Algebraic extension

In field theory, an algebraic extension of a F is a E/F in which every element of E is algebraic over F, meaning that it is a root of some non-constant with coefficients in F. Such extensions are fundamental in , as they capture the structure of fields generated by roots of polynomials over a base field, contrasting with transcendental extensions where elements like \pi or e satisfy no such polynomial equations. A key property is that every finite-degree field extension is algebraic, since elements in a finite-dimensional over F must satisfy a of at most the . Conversely, simple algebraic extensions F(\alpha)/F, generated by a single \alpha, have finite equal to the of the minimal of \alpha over F. For example, \mathbb{Q}(\sqrt{2})/\mathbb{Q} is an algebraic extension of 2, as \sqrt{2} satisfies the x^2 - 2. Infinite algebraic extensions, such as the of \mathbb{Q}, arise when adjoining roots of infinitely many s, and every field admits a unique (up to ) , which is algebraically closed—meaning every non-constant over it splits completely into linear factors. Algebraic extensions play a central role in , where the of a finite measures the symmetries among , enabling solvability by radicals for polynomials. They also underpin , as in the study of fields like cyclotomic extensions, and , where function fields over algebraically closed bases simplify curve and classifications. Subclasses include , where minimal polynomials have distinct , and purely inseparable ones over fields of positive .

Fundamentals

Definition

A field extension is a pair of fields (K, F), where F is a subfield of K, meaning F \subseteq K and both share the same addition and multiplication operations. An element \alpha \in K is algebraic over F if there exists a non-zero f(x) \in F of finite at least 1 such that f(\alpha) = 0. The minimal polynomial of \alpha over F is the of least in F that has \alpha as a , and \alpha is algebraic if and only if this minimal polynomial has finite . A K/F is algebraic if every element \alpha \in K is algebraic over F. In contrast, an element \alpha \in K is transcendental over F if it is not algebraic over F, meaning no non-zero in F has \alpha as a root. A K/F is transcendental if it contains at least one transcendental element over F.

Examples

A fundamental example of an algebraic extension is the field \mathbb{Q}(\sqrt{2}) over \mathbb{Q}, where \sqrt{2} is algebraic over \mathbb{Q} as it satisfies the x^2 - 2 = 0 with rational coefficients. This extension has 2, forming a that adjoins the to \mathbb{Q}. Cyclotomic extensions provide another class of algebraic extensions, such as \mathbb{Q}(\zeta_n)/\mathbb{Q}, where \zeta_n is a primitive n-th satisfying the n-th \Phi_n(x) = 0. For instance, when n = 7, \mathbb{Q}(\zeta_7)/\mathbb{Q} has \phi(7) = 6, where \phi is , and is Galois over \mathbb{Q} with isomorphic to (\mathbb{Z}/7\mathbb{Z})^\times. The field of algebraic numbers \overline{\mathbb{Q}} over \mathbb{Q} exemplifies an infinite algebraic extension, comprising all complex numbers that are roots of nonzero polynomials with rational coefficients. It is the of \mathbb{Q} and thus algebraically closed. Composite extensions illustrate closure under composition: if K/\mathbb{F} and L/\mathbb{F} are algebraic, then the compositum KL/\mathbb{F} is also algebraic. A concrete case is \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}, obtained by adjoining square roots of 2 and 3, which has degree 4 over \mathbb{Q} and basis \{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}. Early examples of algebraic extensions arose from solving equations, with methods dating to ancient civilizations like the Babylonians around 2000 BCE, though the abstract theory of fields and algebraic extensions was formalized in the through works on and number fields.

Properties

Finite algebraic extensions

A finite algebraic extension K/F is one in which the [K : F], defined as the dimension of K as a over F, is finite, say n < \infty. In this case, K is a finite-dimensional over F, and every element \beta \in K satisfies a polynomial equation over F of at most n. This vector space structure underpins many properties, such as the fact that finite extensions are precisely the algebraic extensions that are finitely generated as fields over F. A concrete example is the extension \mathbb{Q}(\sqrt{2})/\mathbb{Q}. The minimal polynomial of \sqrt{2} over \mathbb{Q} is x^2 - 2 = 0, which is irreducible, so [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2. The set \{1, \sqrt{2}\} forms a basis for \mathbb{Q}(\sqrt{2}) as a vector space over \mathbb{Q}, and every element can be uniquely expressed as a + b\sqrt{2} with a, b \in \mathbb{Q}. For towers of finite extensions F \subseteq L \subseteq K, the tower law states that [K : F] = [K : L] \cdot [L : F]. This multiplicativity allows computation of degrees in composite extensions and highlights the hierarchical structure of finite algebraic extensions. When F is a number field with ring of integers \mathcal{O}_F, the integral closure \mathcal{O}_K in a finite algebraic extension K/F consists of all elements of K that are integral over \mathcal{O}_F, and \mathcal{O}_K is finitely generated as an \mathcal{O}_F-module.

Infinite algebraic extensions

An infinite algebraic extension K/F is an algebraic field extension in which the degree [K:F] is infinite. Every element of K is algebraic over F, but there is no finite basis for K as a vector space over F. A key property is that every finite subextension of K/F—that is, the extension generated by any finite subset of K over F—is itself a finite algebraic extension. Such extensions arise naturally as direct limits or inductive limits of chains of finite algebraic subextensions. Specifically, any algebraic extension K/F can be expressed as the union of all its finite subextensions, forming an ascending chain under inclusion. For instance, consider the extension K = \mathbb{Q}(\sqrt{p} \mid p \text{ prime}) over \mathbb{Q}, obtained by adjoining the square roots of all prime numbers; each finite collection of these roots generates a finite extension of \mathbb{Q}, but the full extension is infinite-dimensional. If the base field F is countable, then any algebraic extension K/F is also countable. This holds because the set of monic irreducible polynomials over F is countable, and each such polynomial contributes finitely many roots to K, yielding a countable union of finite sets. The algebraic closure \overline{F} of a field F provides a canonical example of an infinite algebraic extension whenever F is not already algebraically closed, as \overline{F} contains roots for all non-constant polynomials over F but has infinite degree over F. In particular, the field of algebraic numbers \overline{\mathbb{Q}} is a countably infinite algebraic extension of \mathbb{Q}.

Characterizations and Theorems

Primitive element theorem

The primitive element theorem provides a key characterization of simple algebraic extensions and plays a central role in the structure theory of finite field extensions. It states that for a finite algebraic extension K/F of fields, K is a simple extension (i.e., K = F(\alpha) for some \alpha \in K) if and only if there are only finitely many intermediate fields between F and K. One direction follows readily from the fact that if K = F(\alpha), then the intermediate fields correspond to the divisors of the minimal polynomial of \alpha over F, which are finite in number. The converse requires more care and is established through the construction of a primitive element, as detailed below. In the separable case, the theorem yields a stronger affirmative result: every finite separable extension K/F is simple. This version, crucial for applications in , ensures that such extensions can always be generated by a single element, simplifying the study of their automorphisms and subfields. Historically, early forms of the theorem appeared in the work of in 1837, who proved it for extensions obtained by adjoining radicals in the context of ruler-and-compass constructions, demonstrating impossibilities like angle trisection. generalized the result in the 1920s to arbitrary finite separable extensions within her abstract framework for field theory, decoupling it from specific constructions like radicals and embedding it in the modern theory of ideals and modules. A proof sketch for the separable case proceeds by induction on the degree [K:F]. If K = F(\alpha_1, \dots, \alpha_n) with each \alpha_i separable over F, assume the result holds for fewer generators. To combine two elements \alpha and \beta with distinct minimal polynomials over F, consider elements of the form \gamma = \alpha + c\beta for c \in F. Since F is infinite (or handled separately if finite), there are only finitely many values of c that would make intermediate fields coincide improperly. Dedekind's discriminant lemma ensures linear independence of characters (or traces) for distinct embeddings, avoiding these finitely many "bad" c and yielding F(\alpha, \beta) = F(\gamma). Iterating this process constructs a single primitive element \theta such that K = F(\theta). For finite fields, the cyclic nature of the multiplicative group provides a generator directly. As a corollary, every finite separable extension is simple, linking this theorem to the broader theory of separable extensions where characteristic p issues like inseparability are absent. A concrete example is the extension \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}, which has degree 4 and is separable. It equals \mathbb{Q}(\sqrt{2} + \sqrt{3}), where \alpha = \sqrt{2} + \sqrt{3} satisfies the minimal polynomial x^4 - 10x^2 + 1 = 0, obtained by setting x = \sqrt{2} + \sqrt{3}, squaring to x^2 = 5 + 2\sqrt{6}, and squaring again while eliminating radicals (equivalently, (x^2 - 5)^2 - 24 = 0). This illustrates how the theorem reduces a multi-generator extension to a single one, facilitating computations of norms, traces, and Galois groups.

Dedekind's independence theorem

Dedekind's independence theorem asserts that if K/F is a finite of fields and \sigma_1, \dots, \sigma_n are the distinct F-embeddings of K into an algebraic closure \overline{F} of F, then these embeddings, viewed as functions K \to \overline{F}, are linearly independent over K. That is, if \sum_{i=1}^n c_i \sigma_i(x) = 0 for all x \in K with c_i \in K, then c_i = 0 for all i. This theorem was proved by Richard Dedekind in 1871 as part of Supplement XI to the second edition of Dirichlet's Vorlesungen über Zahlentheorie, where he developed a rigorous foundation for Galois theory using field automorphisms and embeddings. Dedekind's work clarified the relationship between field extensions and their automorphism groups, resolving ambiguities in earlier treatments by Galois and others. The proof proceeds by contradiction using the discriminant and properties of the minimal polynomial. Suppose there is a linear dependence relation \sum_{i=1}^n c_i \sigma_i = 0 with not all c_i = 0 and n minimal. Without loss of generality, assume c_1 \neq 0. Let \alpha \in K be a primitive element with minimal polynomial f(T) \in F[T] of degree [K:F] = n. The roots of f are \sigma_1(\alpha), \dots, \sigma_n(\alpha). The relation applied to \alpha gives a linear dependence among these roots over K, but multiplying through by the minimal polynomial of \alpha over F(\sigma_2(\alpha), \dots, \sigma_n(\alpha)) leads to a polynomial over F of degree less than n vanishing at \sigma_1(\alpha), contradicting the minimality of f. Alternatively, the relation implies that the Vandermonde-like matrix formed by the embeddings applied to a power basis has a kernel, making its determinant (related to the discriminant of f) zero, but the discriminant is nonzero since the extension is separable. A key application is to establish that the order of the Galois group \mathrm{Gal}(K/F) equals the degree [K:F]. The Galois group consists precisely of the F-automorphisms of K, which are a subset of the F-embeddings into \overline{F}. Since the number of distinct embeddings equals [K:F] by properties of normal extensions, and the automorphisms are linearly independent over K, there must be exactly [K:F] automorphisms. For example, consider the Galois extension \mathbb{Q}(\zeta_3)/\mathbb{Q}, where \zeta_3 is a primitive cube root of unity satisfying \zeta_3^2 + \zeta_3 + 1 = 0. This extension has degree 2, with Galois group \{ \mathrm{id}, \tau \}, where \tau(\zeta_3) = \zeta_3^2. The two embeddings are \sigma_1 = \mathrm{id} and \sigma_2 = \tau, both into \mathbb{C}. Suppose c_1 \sigma_1(x) + c_2 \sigma_2(x) = 0 for all x \in \mathbb{Q}(\zeta_3) with c_1, c_2 \in \mathbb{Q}(\zeta_3). Applying to 1 gives c_1 + c_2 = 0, and to \zeta_3 gives c_1 \zeta_3 + c_2 \zeta_3^2 = 0. Substituting yields c_1 (\zeta_3 - \zeta_3^2) = 0, and since \zeta_3 \neq \zeta_3^2, c_1 = 0, hence c_2 = 0, confirming independence.

Normal and Separable Extensions

Normal extensions

In field theory, an algebraic extension K/F is defined to be normal if every irreducible polynomial in F that has at least one root in K splits completely into linear factors in K. This condition ensures that K contains all conjugates of its elements over F, making the extension stable under conjugation in the sense that all F-embeddings of K into an algebraic closure of F have the same image. For finite extensions, this definition aligns with the requirement that the minimal polynomial of every element in K over F splits completely in K. An equivalent characterization is that K/F is normal if and only if K is the splitting field over F of some set of polynomials in F. In the finite case, this reduces to a single polynomial, where K is generated by the roots of that polynomial and contains all of them. For infinite algebraic extensions, the set may be infinite, ensuring that every finite subextension is contained in the splitting field of a finite subset of the polynomials. This equivalence underscores the role of normal extensions as minimal fields closed under the roots of specified polynomials over the base field. When K/F is a finite normal extension, it is Galois if and only if it is also separable (see ). In this case, the Galois group \mathrm{Gal}(K/F) consists of all F-automorphisms of K and acts transitively on the roots of any irreducible polynomial over F with a root in K. A representative example is the splitting field of x^3 - 2 over \mathbb{Q}, which is \mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2}, \zeta_3) where \zeta_3 is a primitive cube root of unity; this extension is normal of degree 6 over \mathbb{Q}, as it contains all three roots \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2}, \zeta_3 \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2}, and \zeta_3^2 \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2}.

Separable extensions

In field theory, an element \alpha in an algebraic extension L/K is said to be separable over the base field K if the minimal polynomial of \alpha over K has distinct roots in a splitting field, or equivalently, if the minimal polynomial and its formal derivative are coprime in K[X]. This condition ensures that \alpha does not introduce multiple roots that could lead to inseparability. An algebraic extension L/K is separable if every element of L is separable over K. For finite extensions, separability admits a useful characterization in terms of embeddings: a finite extension L/K is separable if and only if the degree [L:K] equals the number of distinct K-embeddings of L into an algebraic closure of K. This equivalence highlights the role of separability in preserving the full count of automorphisms or homomorphisms, contrasting with inseparable cases where embeddings are fewer due to multiple roots. Inseparable extensions arise exclusively in positive characteristic and involve minimal polynomials with multiple roots; for instance, consider the extension \mathbb{F}_p(t)/\mathbb{F}_p(t^p), where t is transcendental, and the minimal polynomial X^p - t over \mathbb{F}_p(t^p) has a single root of multiplicity p since its derivative is zero. A key result is that every finite extension of a perfect field is separable. A field K is perfect if every algebraic extension is separable, which holds for all fields of characteristic zero and for fields of characteristic p > 0 where every element is a p-th power (i.e., K = K^p); finite fields satisfy this condition. For example, the extension \mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})/\mathbb{Q} is separable, as the minimal polynomial X^3 - 2 has three distinct roots and the extension degree matches the number of embeddings into \mathbb{C}. In contrast, over a field of characteristic 3, adjoining a cube root of 2 yields an inseparable extension because the derivative of X^3 - 2 vanishes, leading to a multiple root if irreducible.

Algebraic Closures

Algebraic closure

An algebraic closure of a field F is an algebraically closed field \Omega containing F as a subfield such that the extension \Omega / F is algebraic. This means every element of \Omega satisfies a polynomial equation with coefficients in F, and \Omega has no proper algebraic extensions. The existence of an algebraic closure follows from Zorn's lemma: consider the partially ordered set of all algebraic field extensions of F, ordered by inclusion; a maximal element in this set is an algebraic closure of F. Alternatively, it can be constructed via transfinite induction by iteratively adjoining roots of polynomials. Any two algebraic closures of F are isomorphic as fields over F, ensuring a unique structure up to isomorphism fixing F. In an algebraic closure \Omega of F, every non-constant polynomial in F splits completely into linear factors. The degree of the extension [\Omega : F] is infinite unless F is already algebraically closed, in which case \Omega = F. A classic example is the field of complex numbers \mathbb{C}, which is an algebraic closure of the real numbers \mathbb{R}; this follows from the , stating that every non-constant with real coefficients has a root in \mathbb{C}, and the extension is algebraic of degree 2./10%3A_Roots_of_Polynomials/10.02%3A_The_Fundamental_Theorem_of_Algebra) Another example is the \overline{\mathbb{Q}} of the rational numbers \mathbb{Q}, consisting of all algebraic numbers, which forms a countably infinite extension.

Relative algebraic closures

In field theory, given fields F \subseteq L, the relative algebraic closure of F in L, often denoted \overline{F}^L or simply the algebraic elements of L over F, is defined as the subfield of L consisting of all elements that are algebraic over F. This construction ensures it is a maximal algebraic extension of F contained within L, containing F and closed under addition and multiplication within L. Key properties include the fact that if L is algebraically closed, then the relative algebraic closure of F in L coincides with the (absolute) algebraic closure of F. In the general case where L may not be algebraically closed, it remains the set of all elements in L algebraic over F, forming an algebraic extension that is invariant under any F-automorphism of L. Relative algebraic closures exist in any field extension L/F by taking the union of all subfields of L algebraic over F, and they are unique as subsets of L. For example, consider F = \mathbb{Q} and L = \mathbb{C}, where \mathbb{C} is algebraically closed. The relative algebraic closure of \mathbb{Q} in \mathbb{C} is \overline{\mathbb{Q}} \cap \mathbb{C} = \overline{\mathbb{Q}}, the field of algebraic numbers. In , relative algebraic closures play a crucial role in analyzing infinite extensions and arguments, particularly for identifying intermediate fields and computing Galois groups over non-normal bases; for instance, if M is an intermediate field in an algebraically closed extension \Omega/F, the relative algebraic closure M^1 of M in \Omega is Galois over M.