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Galois extension

In field theory, a Galois extension is a field extension K/F that is algebraic, normal, and separable, meaning every irreducible polynomial over F with a root in K splits completely into linear factors in K, and the minimal polynomial of every element in K over F has distinct roots.^[1] Equivalently, K is the splitting field over F of a separable polynomial (or a finite collection of separable polynomials).^[1] This structure allows for a deep connection between the arithmetic of the fields and the symmetry captured by the Galois group \mathrm{Gal}(K/F), the group of field automorphisms of K that fix F pointwise.^[1] The theory of Galois extensions originated with the work of Évariste Galois in the early 19th century, who, at the age of 17 in 1829, began developing ideas to determine when polynomial equations are solvable by radicals, building on earlier contributions from mathematicians like Lagrange and Ruffini.^[2] Galois' manuscripts, submitted to the French Academy but initially overlooked, were published posthumously in 1846, laying the groundwork for modern abstract algebra.^[2] The concept was later formalized and popularized by figures such as Joseph Liouville and Camille Jordan in the mid-19th century, with Emil Artin providing a streamlined modern treatment in the 20th century that emphasized the group-theoretic perspective.^[3] A cornerstone of the theory is the Fundamental Theorem of Galois Theory, which states that for a finite Galois extension K/F with Galois group G = \mathrm{Gal}(K/F), there is a bijection between the subgroups of G and the intermediate fields F \subseteq L \subseteq K, given by mapping a subgroup H \leq G to its fixed field K^H = \{ \alpha \in K \mid \sigma(\alpha) = \alpha \ \forall \sigma \in H \}.^[2] This correspondence is inclusion-reversing: larger subgroups correspond to smaller fixed fields, and the degree [K : K^H] = |H|.^[2] Moreover, an intermediate field L is normal over F if and only if \mathrm{Gal}(K/L) is a normal subgroup of G, in which case \mathrm{Gal}(L/F) \cong G / \mathrm{Gal}(K/L).^[2] As a key corollary, |G| = [K : F], linking the order of the Galois group directly to the degree of the extension.^[2] Galois extensions are fundamental in number theory, algebraic geometry, and beyond, enabling the study of solvability of polynomials (e.g., quintics are not generally solvable by radicals due to non-abelian Galois groups), constructions of regular polygons, and inverse Galois problems.^[3] For instance, the extension \mathbb{Q}(i)/\mathbb{Q} is Galois with group \mathbb{Z}/2\mathbb{Z}, generated by complex conjugation, while

\mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{2})/\mathbb{Q}

is not, as it lacks all roots of x^4 - 2.^[1] The theory extends to infinite cases via profinite completions, but finite Galois extensions remain the core focus for most applications.^[4]

Definition and Fundamentals

Definition

In field theory, a field extension E/F is called a Galois extension if it is algebraic, normal, and separable. An extension is algebraic if every element of E satisfies a polynomial equation with coefficients in F; it is normal if every irreducible polynomial in F that has a root in E splits completely into linear factors over E; and it is separable if the minimal polynomial over F of every element in E has distinct roots in an algebraic closure.^[5] An equivalent characterization is that E/F is Galois if and only if the fixed field of the group of F-automorphisms of E, denoted \mathrm{Aut}(E/F) or the Galois group \mathrm{Gal}(E/F), is exactly F. For finite extensions, this condition holds if and only if the order of \mathrm{Gal}(E/F) equals the degree [E:F].^[6] Basic properties of Galois extensions include closure under certain operations: if E_1/F and E_2/F are Galois extensions, then their compositum E_1 E_2 (the smallest field containing both) and intersection E_1 \cap E_2 are also Galois over F. For any finite Galois extension E/F, the degree [E:F] equals |\mathrm{Gal}(E/F)|.^[6]

Historical background

The concept of Galois extensions traces its origins to the pioneering work of Évariste Galois in the early 1830s, when he developed a framework to determine the solvability of polynomial equations by radicals. At the age of 17, Galois submitted his first memoir to the French Academy of Sciences in 1829 outlining conditions under which an algebraic equation could be solved using radicals, building on earlier ideas about permutations of roots but introducing the innovative notion that the structure of associated groups governs solvability.^[7]^[8] His subsequent submissions in 1831 faced rejection due to perceived lack of rigor, yet these efforts laid the foundational groundwork for what would become Galois theory, linking field extensions to group-theoretic properties. Tragically, Galois died on May 31, 1832, at age 20, from wounds sustained in a duel, leaving his ideas largely unpublished and unrecognized during his lifetime.^[7]^[8] The revival of Galois's work came through the efforts of Joseph Liouville, who recognized its significance in 1843 and edited and published Galois's collected mathematical memoirs in his Journal de Mathématiques Pures et Appliquées in 1846. This publication included previously unpublished manuscripts and provided an overview of Galois's contributions, bringing his insights on equation solvability to the broader mathematical community and sparking interest in the theory despite initial obscurity. Liouville's edition marked a crucial step in preserving and disseminating Galois's legacy, influencing subsequent developments in algebra.^[9]^[8]^[10] In the 1920s and 1930s, Emil Artin reformulated Galois theory in a modern abstract algebraic framework, shifting the focus from classical equation solvability to the relationship between field extensions and their automorphism groups. Artin introduced the key concepts of fixed fields—subfields invariant under a group of automorphisms—and demonstrated how these correspond bijectively to subgroups of the Galois group, thereby defining Galois extensions as those where the extension degree equals the order of the automorphism group. This perspective, detailed in his 1938 Grundzüge der Galois'schen Theorie and expanded in his 1942 lectures (published as Galois Theory in 1944), transitioned the theory from its roots in radical solvability to a cornerstone of abstract field theory, emphasizing normality and separability through group structures.^[11]^[12]^[13]

Characterizations

Finite Galois extensions

A finite field extension E/F of degree n is Galois if and only if |\operatorname{Aut}(E/F)| = n.^[6] This condition equivalently means that the fixed field of \operatorname{Aut}(E/F) is precisely F.^[6] Another characterization is that E/F is Galois if and only if every irreducible polynomial in F having at least one root in E splits completely into distinct linear factors in E.^[14] This captures both the normality (complete splitting) and separability (distinct roots) properties inherent to the extension.^[14] Artin's theorem provides a constructive characterization: a finite extension E/F is Galois if and only if E is the splitting field over F of some separable polynomial in F.^[14] Equivalently, by Artin's lemma, if G is a finite group of automorphisms of a field E such that [E : E^G] is finite, then E/E^G is Galois with Galois group exactly G.^[6] For separability in the finite case, an extension E/F is separable if every element of E is separable over F, meaning its minimal polynomial over F has distinct roots; combined with normality, this yields the Galois property.^[14]

Infinite Galois extensions

In the infinite case, a field extension E/F is defined as Galois if it is algebraic, normal, and separable, or equivalently, if E is the union of a directed system of finite Galois subextensions E_i/F such that the automorphism group \mathrm{Aut}(E/F) acts continuously on E with respect to the discrete topology on E and the Krull topology on \mathrm{Aut}(E/F).^[15]^[16] This generalization extends the finite characterizations by allowing unbounded degree while preserving normality and separability through the union structure.^[17] A key characterization is that the fixed field of \mathrm{Aut}(E/F), denoted \mathrm{Fix}(\mathrm{Aut}(E/F)), equals F, and every finite subextension of E/F is itself Galois over F.^[15]^[17] This ensures that the infinite extension inherits the splitting properties of its finite components, with the Galois group \mathrm{Gal}(E/F) = \mathrm{Aut}(E/F) acting faithfully to recover the base field.^[16] The Galois group \mathrm{Gal}(E/F) possesses an inverse limit structure, arising as a profinite group that is the inverse limit \varprojlim \mathrm{Gal}(E_i/F) over the directed system of finite Galois subextensions E_i/F.^[15]^[16]^[17] In this topology, open normal subgroups correspond to the \mathrm{Gal}(E/E_i), yielding finite quotients isomorphic to \mathrm{Gal}(E_i/F).^[15] Unlike finite Galois extensions, where the group order equals the extension degree, infinite Galois extensions lack a direct cardinality equality due to the infinite degree; however, the finite quotients of \mathrm{[Gal}(E/F)](/page/Galois_group) are dense in the profinite topology, capturing the structure through these approximations.^[16]^[17]

Galois Group and Fundamental Theorem

The Galois group

In Galois theory, for a Galois extension E/F of fields, the Galois group, denoted \Gal(E/F) or \Aut_F(E), is defined as the group of all field automorphisms of E that fix every element of F pointwise.^[18] This group captures the symmetries of the extension while preserving the base field structure.^[19] When E/F is a finite Galois extension, \Gal(E/F) is a finite group whose order equals the degree of the extension [E:F].^[20] In contrast, for infinite Galois extensions, \Gal(E/F) is endowed with the Krull topology, defined such that the basic open neighborhoods of the identity consist of cosets of subgroups fixing finite Galois subextensions; this topology renders \Gal(E/F) a profinite group, which is compact, Hausdorff, and totally disconnected.^[4] The Galois group acts faithfully on the roots of any separable irreducible polynomial over F whose splitting field is contained in E, permuting them while respecting the field relations.^[21] In the finite case, this action embeds \Gal(E/F) as a transitive subgroup of the symmetric group on the set of roots, reflecting the extension's minimal polynomial degree.^[22] Subgroups of \Gal(E/F) are in one-to-one correspondence with intermediate fields between F and E via the fixed-field map, where the fixed field of a subgroup H is \{x \in E \mid \sigma(x) = x \ \forall \sigma \in H\}, providing the lattice structure central to Galois theory.^[19]

Fundamental theorem of Galois theory

The fundamental theorem of Galois theory establishes a profound connection between the subfields of a Galois extension and the subgroups of its Galois group, providing a bijective correspondence that reveals the structural symmetries of field extensions. For a finite Galois extension E/F with Galois group G = \Gal(E/F), there is a bijection between the set of intermediate fields K such that F \subseteq K \subseteq E and the set of subgroups H \leq G. This bijection is defined by associating to each subgroup H its fixed field E^H = \{ x \in E \mid \sigma(x) = x \ \forall \sigma \in H \}, and to each intermediate field K the subgroup \Gal(E/K) = \{ \sigma \in G \mid \sigma(k) = k \ \forall k \in K \}. These maps are inverses: \Gal(E/E^H) = H and (E^{\Gal(E/K)}) = K.^[14]^[23] Moreover, the degrees satisfy [E : K] = |H| where H = \Gal(E/K), and [K : F] = [G : H], reflecting the index of the subgroup in the full Galois group. Normal subgroups H \triangleleft G correspond precisely to those intermediate fields K for which K/F is itself Galois, with the isomorphism \Gal(K/F) \simeq G/H. This correspondence is order-reversing: if H_1 \supseteq H_2, then E^{H_1} \subseteq E^{H_2}, and conversely for fields.^[14]^[23] In the infinite case, for an algebraic Galois extension \Omega/F with Galois group G = \Gal(\Omega/F) equipped with the Krull topology (making G a profinite group), the bijection extends to closed subgroups H of G and intermediate fields M with F \subseteq M \subseteq \Omega. The maps are again H \mapsto \Omega^H and M \mapsto \Gal(\Omega/M), which are continuous and inverses on closed subgroups. Open subgroups correspond to finite Galois subextensions over F. Normal closed subgroups yield Galois intermediate fields, preserving the quotient isomorphism.^[14] The proof for the finite case relies on the separability and normality of the extension, ensuring that the number of F-embeddings of E into an algebraic closure equals [E : F] = |G|, and that fixed fields are precisely the intermediates fixed by subgroups via Artin's lemma, which shows E/E^H is Galois with group H. Injectivity follows from distinct fixed fields implying distinct subgroups, and surjectivity from the fixed field of \Gal(E/K) recovering K. For the infinite case, the Krull topology ensures closure under limits, with Zorn's lemma applied to chains of closed subgroups to handle arbitrary intermediates, and continuity of restriction homomorphisms preserves the correspondence.^[14]^[23] As an implication, the theorem induces a lattice anti-isomorphism between the lattice of subgroups of G (ordered by inclusion) and the lattice of intermediate fields (ordered by inclusion), enabling the translation of field-theoretic questions into group-theoretic ones and vice versa. This duality underpins much of modern Galois theory, facilitating the study of extension structures through symmetry groups.^[14]

Examples

Finite examples

One of the simplest finite Galois extensions is the quadratic extension \mathbb{Q}(\sqrt{2})/\mathbb{Q}, which has degree 2 and is the splitting field of the irreducible polynomial x^2 - 2 over \mathbb{Q}.^[24] The minimal polynomial of \sqrt{2} over \mathbb{Q} is x^2 - 2, and adjoining \sqrt{2} yields both roots \pm \sqrt{2}, making the extension normal and separable. The Galois group \mathrm{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) is cyclic of order 2, isomorphic to \mathbb{Z}/2\mathbb{Z}, generated by the automorphism \sigma: \sqrt{2} \mapsto -\sqrt{2}.^[24] A contrasting cubic example illustrates the distinction between separable extensions and Galois extensions. The extension

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

has degree 3 and is separable, as the minimal polynomial x^3 - 2 is separable over \mathbb{Q}, but it is not normal because it contains only one real root

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

and misses the two complex roots

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

and

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

, where \omega = e^{2\pi i / 3} is a primitive cube root of unity.^[25] In contrast, the splitting field of x^3 - 2 over \mathbb{Q} is

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

, which has degree 6 over \mathbb{Q} and is a Galois extension with Galois group isomorphic to S_3, the symmetric group on 3 letters.^[26] This group is non-abelian, reflecting the extension's structure as a compositum of the degree-3 extension

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

and the quadratic extension \mathbb{Q}(\omega)/\mathbb{Q}. Another explicit example of a degree-6 Galois extension with non-abelian Galois group S_3 is the splitting field of the irreducible cubic polynomial x^3 + x^2 - 2x - 1 over \mathbb{Q}. This polynomial has three distinct real roots, and its splitting field over \mathbb{Q} has degree 6, confirming it is Galois with Galois group S_3.^[27] The group action includes 3-cycles corresponding to permutations of the roots and transpositions arising from quadratic subextensions. Extensions of finite fields provide further finite Galois examples. For a prime p and integer n \geq 1, the extension \mathrm{GF}(p^n)/\mathrm{GF}(p) is always Galois of degree n, as \mathrm{GF}(p^n) is the splitting field of the irreducible polynomial x^{p^n} - x over \mathrm{GF}(p).^[28] The Galois group \mathrm{Gal}(\mathrm{GF}(p^n)/\mathrm{GF}(p)) is cyclic of order n, generated by the Frobenius automorphism \phi: x \mapsto x^p.^[28] This structure holds for all such extensions, making finite field extensions a canonical class of abelian Galois extensions.

Infinite examples

Infinite Galois extensions arise as unions of ascending chains of finite Galois extensions, providing examples where the Galois group is a profinite group rather than finite.^[29] These structures illustrate how infinite extensions capture the full scope of Galois theory in number fields and local fields. A prominent example is the infinite cyclotomic extension \mathbb{Q}(\zeta_\infty)/\mathbb{Q}, where \zeta_\infty denotes the union of all roots of unity over \mathbb{Q}. This extension is Galois, obtained as the direct limit of the finite cyclotomic extensions \mathbb{Q}(\zeta_n)/\mathbb{Q} for n \in \mathbb{N}, each of which is Galois with group (\mathbb{Z}/n\mathbb{Z})^\times. The Galois group \mathrm{Gal}(\mathbb{Q}(\zeta_\infty)/\mathbb{Q}) is the profinite completion of \mathbb{Z}^\times, isomorphic to \prod_p \mathbb{Z}_p^\times, where the product runs over all primes p. This group acts continuously on the roots of unity, reflecting the inverse limit structure over the finite cases.^[29] Another key example is the algebraic closure \overline{\mathbb{Q}}/\mathbb{Q}, the maximal algebraic extension of the rationals. This infinite Galois extension has absolute Galois group \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), a profinite group of enormous complexity that encodes all finite Galois extensions of \mathbb{Q}. Unlike the cyclotomic case, this group is not abelian and admits no simple explicit description, though its profinite topology arises from the inverse system of all finite quotients corresponding to finite Galois extensions. Subgroups correspond to intermediate fields via the fundamental theorem, but the full group's structure remains a central object in modern number theory, with open questions about its profinite presentations. In the local setting, consider the maximal unramified extension \mathbb{Q}_p^{\mathrm{unr}}/\mathbb{Q}_p of the p-adic rationals. This extension is Galois, formed as the union of all finite unramified extensions, each isomorphic to adjoining roots of unity of order p^k - 1 in the residue field. The Galois group \mathrm{Gal}(\mathbb{Q}_p^{\mathrm{unr}}/\mathbb{Q}_p) is isomorphic to \hat{\mathbb{Z}}, the profinite completion of \mathbb{Z}, generated topologically by the Frobenius automorphism that acts as the p-power map on residues. This group is procyclic, highlighting the simplicity of unramified extensions in local fields compared to ramified ones.^[30] An example of an infinite radical extension that is Galois arises in the p-adic Kummer tower, such as \mathbb{Q}_p(\zeta_{p^\infty}, a^{1/p^\infty})/\mathbb{Q}_p(\zeta_{p^\infty}) for a p-adic unit a not a p-th power, where \zeta_{p^\infty} are the p-power roots of unity. This extension is obtained by iteratively adjoining p^n-th roots of a and is Galois with group \mathbb{Z}_p, the p-adic integers, acting via multiplication on the roots. The full tower over \mathbb{Q}_p requires the roots of unity for normality, yielding a pro-p Galois group that demonstrates solvability by radicals in the infinite case through Kummer theory.^[29]

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