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Galois group

In mathematics, the Galois group of a E/F is defined as the group of all field automorphisms of E that fix every element of the base field F pointwise, with the group operation given by composition of automorphisms. This structure was introduced by the French mathematician (1811–1832), whose pioneering work laid the foundations of modern by linking to the solvability of equations by radicals. Galois groups are particularly studied in the context of Galois theory, where they encode essential information about the symmetries of field extensions and determine whether certain extensions are solvable in terms of radicals. A field extension E/F is called Galois if it is normal and separable, in which case the order of the Galois group \mathrm{Gal}(E/F) equals the degree [E:F], providing a precise measure of the extension's complexity. For a polynomial f(x) \in F, the Galois group is typically defined as \mathrm{Gal}(K/F), where K is the splitting field of f over F, and it acts faithfully as a permutation group on the roots of f, embedding into the symmetric group S_n with n = \deg(f). If f is irreducible over F, this action is transitive, meaning the group permutes the roots without fixed points among them, and the group's order is divisible by n. The establishes a bijective between the s of \mathrm{Gal}(E/F) and the intermediate fields F \subseteq M \subseteq E, where the fixed field of a H is the set of elements in E fixed by every in H, and conversely, the corresponding to an intermediate field M consists of automorphisms fixing M. This theorem implies that normal subgroups correspond to normal extensions and quotient groups to quotient fields, enabling the use of group-theoretic tools to analyze algebraic structures. For instance, the Galois group of the cyclotomic extension \mathbb{Q}(\zeta_p)/\mathbb{Q}, where \zeta_p is a p-th and p is prime, is cyclic of order p-1. Galois groups have profound applications beyond solvability criteria, such as for studying inverse Galois problems (determining which finite groups arise as Galois groups over \mathbb{Q}) for resolving classical construction problems like or circle squaring using . They also appear advanced contexts, including the of Galois representations , where the \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) encodes arithmetic data. Historically, Galois's ideas, developed amid political turmoil , revolutionized algebra by shifting focus from explicit solutions to structural invariants, influencing fields from to physics.

Definition and Basic Concepts

Field extensions and automorphisms

A field extension K/F consists of two fields F and K such that F is a subfield of K. In this setup, K forms a over F, and the of this vector space, denoted [K:F], is called the degree of the extension. The extension is finite if [K:F] is finite. An element \alpha \in K is algebraic over F if it is a root of some nonzero polynomial with coefficients in F. The extension K/F is algebraic if every element of K is algebraic over F. A finite extension is separable if the minimal polynomial of every element over F has distinct roots in a splitting field. It is normal if every irreducible polynomial in F that has a root in K splits completely into linear factors in K. The \operatorname{Aut}(K/F) of the extension K/F is the set of all field automorphisms \sigma: K \to K that fix every of F , forming a group under composition. These automorphisms preserve the operations and the inclusion of F in K. In contrast, the full \operatorname{Aut}(K) consists of all field automorphisms of K without the fixing condition on F, which can be significantly larger for certain . A classic example is the extension \mathbb{C}/\mathbb{R}, where the non-identity automorphism is complex conjugation, defined by \sigma(a + bi) = a - bi for a, b \in \mathbb{R}, which fixes \mathbb{R} and swaps i with -i. Thus, \operatorname{Aut}(\mathbb{C}/\mathbb{R}) has order 2, generated by this conjugation.

Galois group of a field extension

In the context of field extensions, a Galois extension K/F is defined as a field extension that is algebraic, separable, and . For such an extension, the \Gal(K/F) is the group of all field automorphisms of K that fix F , denoted \Aut_F(K) or simply \Aut(K/F). This group captures the symmetries of the extension, where each automorphism is determined by its action on a basis of K over F, preserving the field structure and the relations imposed by elements of F. A fundamental property of finite Galois extensions is that the order of the Galois group equals the degree of the extension: |\Gal(K/F)| = [K : F]. This equality arises because the automorphisms form a basis for the of F-linear maps from K to itself under certain conditions, ensuring the group acts faithfully and transitively on the roots. In particular, elements of \Gal(K/F) permute the roots of any over F that splits in K, acting as permutations on the set of roots while preserving the minimal relations. For instance, if f(x) \in F is irreducible with roots \alpha_1, \dots, \alpha_n in K, then any \sigma \in \Gal(K/F) maps \alpha_i to another root \alpha_j, inducing a transitive action on the roots. The concept of the Galois group originated in Évariste Galois's investigations into the solvability of polynomial equations by radicals, where he sought to characterize when roots could be expressed using nested radicals over the base field. Galois realized that the structure of the of the provides a precise criterion: solvability by radicals corresponds to the group being solvable, linking algebraic symmetries directly to constructibility via radicals. This perspective transformed the study of equations from solutions to a systematic group-theoretic framework.

Galois group of a polynomial

The Galois group of a polynomial f(x) \in F, denoted \Gal(f/F), is defined to be the Galois group \Gal(K/F) of the splitting field K of f over the base field F. The splitting field K is the smallest extension of F that contains all the roots of f. For the extension K/F to be Galois, the polynomial f must be separable, meaning it has distinct roots in an algebraic closure of F (equivalently, f and its formal derivative f' have no common roots). In this case, K/F is both normal and separable, ensuring that \Gal(K/F) captures the full symmetry of the extension. If f is inseparable (which occurs only in positive characteristic), the extension K/F is normal but not separable; in such situations, the Galois group is still defined as the automorphism group \Aut(K/F), though the extension lacks the Galois property, and one may instead study the Galois group of the maximal separable subextension or the separable closure of K over F. The elements of \Gal(f/F) are F-automorphisms of K that permute the roots of f while fixing F. When f is separable of degree n, this action induces a faithful permutation representation of \Gal(f/F) on the n distinct roots, yielding a monomorphism \Gal(f/F) \hookrightarrow S_n into the symmetric group on n letters. For example, the polynomial x^2 - 2 \in \mathbb{Q} is separable, with splitting field \mathbb{Q}(\sqrt{2}) over \mathbb{Q}. The Galois group \Gal(x^2 - 2/\mathbb{Q}) is isomorphic to \mathbb{Z}/2\mathbb{Z}, generated by the automorphism \sqrt{2} \mapsto -\sqrt{2}.

Fundamental Theorem of Galois Theory

Statement of the theorem

The fundamental theorem of Galois theory establishes a one-to-one correspondence between the subgroups of the Galois group of a finite Galois extension and the intermediate fields of that extension. Specifically, let K/F be a finite Galois extension of fields, with Galois group G = \Gal(K/F). Then there exists a bijection between the set of all subgroups H \leq G and the set of all intermediate fields L such that F \subseteq L \subseteq K, given by mapping each subgroup H to its fixed field K^H = \{ x \in K \mid \sigma(x) = x \ \forall \sigma \in H \} and each intermediate field L to \Gal(K/L). This is an anti-isomorphism of posets: if L = K^H, then \Gal(K/L) \cong H, and the G/H \cong \Gal(L/F)./23:_Galois_Theory/23.02:_The_Fundamental_Theorem) The theorem applies precisely when the extension K/F is finite and Galois (i.e., and separable)./23:_Galois_Theory/23.02:_The_Fundamental_Theorem) This result originates from the insights of in the 1830s, which were posthumously developed and rigorously formalized by later mathematicians including and Camille Jordan./23:_Galois_Theory/23.02:_The_Fundamental_Theorem)

Proof sketch and key ideas

The proof of the Fundamental Theorem of Galois Theory, which establishes a bijective correspondence between the intermediate fields of a finite Galois extension K/F and the subgroups of its Galois group \Gal(K/F), proceeds under the assumption that K/F is finite, normal, and separable; extensions to infinite Galois extensions require considering closed subgroups in the Krull topology. A central idea is to invoke the primitive element theorem, which asserts that every finite separable extension admits a primitive element \alpha such that K = F(\alpha). Under this representation, every automorphism \sigma \in \Gal(K/F) is uniquely determined by the image \sigma(\alpha), which must be a root of the minimal polynomial of \alpha over F. This simplifies the analysis of the group action and facilitates the mapping between subgroups and fixed fields. Dedekind's independence lemma plays a pivotal role, stating that if \sigma_1, \dots, \sigma_m are distinct automorphisms in \Gal(K/F), then these automorphisms are linearly independent over K as functions from K to itself. Formally, if \sum_{i=1}^m c_i \sigma_i(x) = 0 for all x \in K with c_i \in K, then all c_i = 0. This , originally due to Dedekind in his supplements to Dirichlet's lectures on , ensures that the dimension of the vector space of automorphisms over the fixed field aligns with the group order, proving |\Gal(K/F)| = [K:F] and bounding the size of fixed fields. The proof unfolds in key steps to verify the bijection. First, for a subgroup H \leq \Gal(K/F), the fixed field K^H = \{ x \in K \mid \sigma(x) = x \ \forall \sigma \in H \} is clearly fixed pointwise by H. To show \Gal(K / K^H) = H, injectivity follows from the fact that any \tau \in \Gal(K / K^H) restricts to an element of H on a basis, using linear independence to exclude extraneous automorphisms; surjectivity then holds by counting dimensions via Dedekind's lemma, ensuring no larger group fixes K^H. For surjectivity of the correspondence, given an intermediate field M with F \subseteq M \subseteq K, the subgroup \Gal(K/M) fixes M by definition, and K^{\Gal(K/M)} = M is established by degree equality [K:M] = |\Gal(K/M)|, again leveraging the primitive element representation and independence to confirm the fixed field coincides with M. These steps confirm the maps H \mapsto K^H and M \mapsto \Gal(K/M) are mutual inverses. For infinite Galois extensions, the theorem generalizes by restricting to the closed subgroups of \Gal(K/F) equipped with the Krull topology, where closure ensures and the fixed field preserves the structure, though the proof requires additional topological arguments beyond the finite case.

Corollaries on fixed fields and normality

A key corollary of the states that for a finite K/F with Galois group G = \Gal(K/F), the extension K/F is if and only if G acts transitively on the roots of every over F that has at least one root in K. Equivalently, K/F is if and only if every in F with a root in K splits completely in K. This characterization highlights the connection between the and the splitting behavior of polynomials, ensuring that captures the full embedding of roots under automorphisms. The fixed field correspondence provides further insights: for any subgroup H \leq G, the fixed field K^H = \{ \alpha \in K \mid \sigma(\alpha) = \alpha \ \forall \sigma \in H \} satisfies [K : K^H] = |H|, establishing a precise degree relation between the extension degree and the subgroup order. Moreover, the intermediate extension K^H / F is Galois if and only if H is a normal subgroup of G. This normality condition ensures that the quotient group G/H corresponds to the Galois group of the fixed field extension, preserving the Galois structure in the lattice. The bijection between subgroups of G and intermediate fields, which is inclusion-reversing, underpins these relations. For separable extensions, every finite separable extension L/F admits a Galois closure, which is a finite M/F containing L such that M is the over F of the product of the minimal polynomials of a basis for L over F. This closure exists because separability ensures the existence of enough automorphisms to embed the extension into a one, facilitating the study of non-normal separable extensions via their Galois hulls. In the infinite Galois theory setting, where K/F is an infinite algebraic Galois extension, the fundamental theorem extends via the Krull topology on \Gal(K/F), which equips the group with a profinite structure to maintain the correspondence between closed subgroups and intermediate fixed fields. This topology, defined by basic open sets as cosets of open subgroups corresponding to finite Galois subextensions, ensures compactness and allows the theory to handle infinite degrees through profinite completions.

Structure of Galois Groups

Lattice of subgroups and intermediate fields

The fundamental theorem of Galois theory establishes an anti-isomorphism between the lattice of subgroups of the Galois group G = \mathrm{Gal}(K/F) of a Galois extension K/F and the lattice of intermediate fields F \subseteq L \subseteq K, ordered by inclusion. This correspondence maps each subgroup H \leq G to its fixed field K^H = \{ x \in K \mid \sigma(x) = x \ \forall \sigma \in H \}, and each intermediate field L to the subgroup \mathrm{Gal}(K/L) = \{ \sigma \in G \mid \sigma(l) = l \ \forall l \in L \}; these maps are inverses, yielding a bijection that reverses the order. Under this , the relations are preserved in reverse: for subgroups H_1, H_2 \leq G, H_1 \leq H_2 K^{H_2} \leq K^{H_1}. In particular, H \leq G implies K^H \geq K^G = F, the base field fixed by the full group. This structure aligns with the fixed field correspondence outlined in prior corollaries, where the fixed field of a fully determines the intermediate extensions. The structure is often visualized using Hasse diagrams, which depict the partial order by inclusion without transitive edges. For a extension (degree 2), the field is a F < K, corresponding to the subgroup \{e\} < G (order reversed), forming a simple two-level diagram. For the splitting field of an irreducible cubic polynomial over F with Galois group S_3 (degree 6 extension), the diagrams show branching: the field has a intermediate field above F and three cubic intermediate fields above F, all below K; the subgroup mirrors this inversely and includes the normal subgroup A_3 of order 3 and three order-2 subgroups. These diagrams illustrate the completeness of the poset in finite cases, where every subgroup and intermediate field appears exactly once in the bijection, forming a complete under joins (generated fixed fields) and meets (intersections of subgroups). In the finite Galois case, the correspondence is fully bijective and captures all intermediate structures without gaps. For infinite extensions, the lattice requires additional topology on the Galois group (as a profinite group), restricting to closed subgroups to ensure the fixed fields yield all intermediate extensions and maintain the anti-isomorphism.

Normal subgroups and quotient groups

In the context of Galois theory, a subgroup H of the Galois group G = \Gal(K/F) of a Galois extension K/F is normal if and only if the fixed field L = K^H is a Galois extension of the base field F. This correspondence arises from the fundamental theorem of Galois theory, where the normality of H ensures that every automorphism in G maps L to itself, making L/F normal and separable. Under this condition, the Galois group \Gal(L/F) is isomorphic to the quotient group G/H, established via the first isomorphism theorem applied to the restriction homomorphism \phi: G \to \Aut(L/F) defined by \phi(\sigma) = \sigma|_L, whose kernel is precisely H. Additionally, the subgroup \Gal(K/L) is isomorphic to H, reflecting how the quotient action factors through the intermediate extension. This structure preserves the group-theoretic properties while mirroring the tower of field extensions. A concrete illustration occurs when G is cyclic, as in the Galois group of a cyclotomic extension \mathbb{Q}(\zeta_p)/\mathbb{Q} for prime p, which is cyclic of order p-1 and thus abelian. In such cases, every subgroup of G is normal, so all intermediate fixed fields are Galois over \mathbb{Q}, with quotients also cyclic. The presence of normal subgroups enables the construction of solvable series for G, consisting of chains \{e\} = H_0 \trianglelefteq H_1 \trianglelefteq \cdots \trianglelefteq H_n = G where each quotient H_{i+1}/H_i is abelian; such series are central to analyzing the solvability of the Galois group in abstract terms.

Solvability and radical extensions

A Galois group is solvable if it admits a composition series whose successive factor groups are abelian. This structural property plays a central role in determining whether a polynomial equation can be solved using radicals. Specifically, for an irreducible polynomial f(x) over a field F of characteristic zero, f(x) is solvable by radicals if and only if its Galois group \mathrm{Gal}(f) is solvable. This equivalence is encapsulated in Galois' theorem, which establishes a profound link between the algebraic structure of the Galois group and the expressibility of roots via nested radicals. In characteristic zero, the theorem asserts that the roots of f(x) lie in a radical extension of F precisely when \mathrm{Gal}(f) possesses a subnormal series with abelian quotients. This criterion was originally developed by Évariste Galois in his 1831 memoir, providing a necessary and sufficient condition for solvability. A radical extension of a field F is constructed as a tower F = F_0 \subset F_1 \subset \cdots \subset F_m where each F_i = F_{i-1}(\alpha_i) with \alpha_i^{n_i} \in F_{i-1} for some integer n_i \geq 2. Such extensions correspond to solvable Galois groups because each step adjoins an n_i-th root, yielding a cyclic quotient in the associated Galois correspondence when the base field contains the necessary roots of unity. The abelian nature of these quotients ensures that the full Galois group of the extension is solvable, mirroring the composition series with abelian factors. A classic counterexample arises with quintic polynomials whose Galois group is the alternating group A_5, which is simple and non-abelian, hence unsolvable. Since A_5 lacks a composition series with abelian factors, any irreducible quintic over \mathbb{Q} with this Galois group cannot be solved by radicals, demonstrating the sharpness of Galois' criterion.

Properties of Galois Groups

Order and degree relations

In Galois theory, for a finite Galois extension K/F of fields, the order of the Galois group \lvert \mathrm{Gal}(K/F) \rvert equals the degree of the extension [K : F]. This equality arises from the fundamental theorem of Galois theory, which establishes a bijective correspondence between the subgroups of \mathrm{Gal}(K/F) and the intermediate fields between F and K, with the index of a subgroup matching the degree of the corresponding fixed field extension. For a separable polynomial f(x) \in F of degree n, the Galois group \mathrm{Gal}(f) is defined as the Galois group of the splitting field K of f over F, so \lvert \mathrm{Gal}(f) \rvert = [K : F]. Since \mathrm{Gal}(f) acts faithfully as a permutation group on the n roots of f, it embeds as a transitive subgroup of the symmetric group S_n, implying that \lvert \mathrm{Gal}(f) \rvert divides n!. The precise order thus provides the minimal degree of the splitting field extension. The order of the Galois group also influences the factorization behavior of the polynomial over F through invariants like the discriminant and resolvents. For instance, the square-freeness of the discriminant in F determines whether the Galois group lies in the alternating group A_n, which in turn affects whether the polynomial factors into even or odd permutations of roots, corresponding to specific factorization types such as irreducible or products of quadratics. Resolvents, such as the cubic resolvent for quartics, further refine this by testing membership in subgroups like the dihedral or Klein four-group, linking the group's order and structure to the degrees of irreducible factors over F. In the case of infinite Galois extensions, such as the algebraic closure \overline{F}/F, the Galois group \mathrm{Gal}(\overline{F}/F) is infinite and equipped with the Krull topology, making it a profinite group expressed as the inverse limit of the Galois groups of finite Galois subextensions. Here, the "order" is interpreted topologically, with the group as a compact, totally disconnected space whose finite quotients capture the structure of finite extensions, generalizing the finite case via projective limits.

Transitivity and primitivity

The Galois group of a separable polynomial f \in F of degree n acts faithfully as a permutation group on the set of its n roots in the splitting field L of f over F, embedding it as a subgroup G of the S_n. This permutation representation captures the symmetries of the roots under field automorphisms. A key property is transitivity: the action is transitive, meaning G has a single orbit on the roots, if and only if f is irreducible over F. Transitivity ensures that all roots are Galois conjugates of one another, reflecting the irreducibility; the minimal polynomial of any root \alpha has degree n precisely when the orbit size is n, by the applied to G. Beyond transitivity, the action may exhibit primitivity, a stronger condition indicating that the permutation representation has no nontrivial system of imprimitivity. A transitive action of G \leq S_n is primitive if and only if there are no blocks—nontrivial partitions of the roots preserved by G—or equivalently, if the stabilizer H of any root (a subgroup isomorphic to \mathrm{Gal}(L/F(\alpha)), where \alpha is a root) is a maximal subgroup of G. In the Galois setting, this corresponds to the absence of intermediate fields strictly between F(\alpha) and L; the extension L/F(\alpha) has no proper subextensions, making F(\alpha)/F "maximal" in the lattice of subfields. Imprimitive actions arise, for example, when the roots can be grouped into equal-sized subsets (blocks) invariant under G, often linked to structures in the group. Primitive actions have significant implications for the structure of G. For instance, if a primitive permutation group of degree n \geq 3 contains a 3-cycle, then it contains the alternating group A_n as a subgroup. More generally, primitive groups of degree n often contain either A_n or S_n itself, particularly when n is prime or the group is highly symmetric, though exceptions exist for small n or affine types. In the Galois context, this restricts possible realizations: not all abstract primitive groups embed as Galois groups over fields like \mathbb{Q}, but those that do (e.g., S_n for septic polynomials) highlight connections to inverse Galois theory. The classification of primitive permutation groups, via the O'Nan-Scott theorem, divides them into affine, almost simple, and product action types, with Galois realizations favoring almost simple cases like projective linear groups over finite fields.

Frobenius elements in number fields

In the context of a finite Galois extension K/\mathbb{Q} with Galois group G = \mathrm{Gal}(K/\mathbb{Q}), the Frobenius element associated to an unramified prime p of \mathbb{Z} is an element \sigma_p \in G that generates the decomposition group at any prime ideal \mathfrak{P} of the ring of integers \mathcal{O}_K lying above p, and it acts on \mathcal{O}_K by satisfying \sigma_p(x) \equiv x^p \pmod{\mathfrak{P}} for all x \in \mathcal{O}_K. These elements form a conjugacy class in G, independent of the choice of \mathfrak{P} above p, and their existence follows from lifting the Frobenius automorphism x \mapsto x^p on the residue field \mathbb{F}_p. The Frobenius elements play a central role in class field theory through Artin reciprocity, which establishes a canonical homomorphism, known as the Artin map, from the group of ideals of \mathbb{Q} coprime to a finite set of ramified primes to G (in the abelian case), sending the prime ideal (p) to the conjugacy class of \sigma_p. This map is surjective onto G for abelian extensions, linking the arithmetic of ideals in the base field to the structure of the Galois group, and it generalizes quadratic reciprocity to higher-degree extensions. A concrete illustration occurs in cyclotomic fields: for the p-th cyclotomic extension \mathbb{Q}(\zeta_p)/\mathbb{Q} where \zeta_p is a primitive p-th root of unity and p is an odd prime, the Galois group is isomorphic to (\mathbb{Z}/p\mathbb{Z})^\times, and for an unramified prime q \neq p, the Frobenius element \sigma_q is the unique automorphism sending \zeta_p \mapsto \zeta_p^q, thereby generating the full cyclic group of order p-1. The distribution of these Frobenius conjugacy classes among the primes is governed by Chebotarev's density theorem, which asserts that for a fixed conjugacy class C in G, the set of unramified primes p such that the Frobenius class \{\sigma_p\} equals C has natural density |C|/|G| among all primes. This equidistribution result quantifies the arithmetic significance of Frobenius elements, enabling applications to prime splitting and the effective computation of class numbers in number fields.

Examples of Galois Groups

Trivial and cyclic groups

The Galois group of an extension K/F is trivial, consisting solely of the identity automorphism, if and only if K = F. In this case, the extension is Galois of 1, with no non-trivial automorphisms. For extensions in characteristic p > 0, purely inseparable extensions also yield a trivial , as they lack non-trivial separable automorphisms, though such extensions are not Galois unless trivial. By the , the order of the Galois group equals the degree of the extension. Cyclic Galois groups arise in cyclic extensions, where the Galois group is isomorphic to a \mathbb{Z}/n\mathbb{Z} for some n. A prominent example occurs in cyclotomic extensions: for a prime p, the Galois group \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}), where \zeta_p is a p-th , is cyclic of order p-1 and isomorphic to (\mathbb{Z}/p\mathbb{Z})^\times. This isomorphism follows from the action of automorphisms sending \zeta_p to \zeta_p^k for k coprime to p, generating the modulo p. Kummer theory provides a for constructing cyclic extensions via radicals when the base field contains the necessary . Specifically, if K contains a primitive n-th and has coprime to n, then adjoining an n-th root b of some a \in K^\times yields a L = K(b)/K whose is cyclic of order dividing n. The group embeds injectively into the cyclic group \mu_n(K) of n-th roots of unity via the map \sigma \mapsto \sigma(b)/b. In characteristic p > 0, cyclic extensions of degree p are classified by Artin-Schreier theory. An Artin-Schreier extension L/K is obtained by adjoining a root b of x^p - x - a = 0 for a \in K such that the polynomial is irreducible; this extension is Galois with cyclic Galois group of order p, isomorphic to \mathbb{Z}/p\mathbb{Z}. The is determined by \sigma(b) = b + m for m \in \mathbb{Z}/p\mathbb{Z}, reflecting the additive structure in characteristic p.

Abelian groups from cyclotomic and quadratic extensions

Abelian Galois groups arise naturally in certain algebraic number field extensions of the rationals, particularly those that are quadratic or cyclotomic. These extensions provide concrete examples where the Galois group is finite and abelian, illustrating fundamental aspects of Galois theory over \mathbb{Q}. Quadratic extensions of \mathbb{Q} offer the simplest non-trivial abelian Galois groups. For a square-free integer d \neq 1, the extension \mathbb{Q}(\sqrt{d})/\mathbb{Q} is Galois with degree 2, and its Galois group is generated by the automorphism \sigma: \sqrt{d} \mapsto -\sqrt{d}, which is the complex conjugation if d < 0 or the analogous sign change otherwise. Thus, \mathrm{Gal}(\mathbb{Q}(\sqrt{d})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}. This isomorphism holds because the minimal polynomial x^2 - d splits completely in the extension, and the only non-identity automorphism swaps the roots \pm \sqrt{d}. Such groups are cyclic of prime order, emphasizing the abelian nature inherent to quadratic fields. Cyclotomic extensions yield more varied abelian Galois groups. The nth cyclotomic extension \mathbb{Q}(\zeta_n)/\mathbb{Q}, where \zeta_n = e^{2\pi i / n} is a primitive nth , is Galois of degree \varphi(n), with \varphi denoting . The Galois group consists of automorphisms \sigma_k: \zeta_n \mapsto \zeta_n^k for k coprime to n, yielding the explicit \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times. Since (\mathbb{Z}/n\mathbb{Z})^\times is always abelian, every cyclotomic extension over \mathbb{Q} has an abelian Galois group, regardless of n. For prime n = p, this group is cyclic of order p-1, but in general, it can be a product of cyclic groups reflecting the prime factorization of n. The Kronecker-Weber theorem unifies these abelian structures by asserting that every finite abelian extension of \mathbb{Q} is contained within some cyclotomic extension \mathbb{Q}(\zeta_m)/\mathbb{Q}. This result implies that quadratic extensions, being abelian, embed into cyclotomics; for instance, \mathbb{Q}(\sqrt{-1}) \subseteq \mathbb{Q}(\zeta_4) and \mathbb{Q}(\sqrt{5}) \subseteq \mathbb{Q}(\zeta_5). Consequently, all abelian Galois groups over \mathbb{Q} can be realized as quotients of units modulo m, highlighting the centrality of cyclotomic fields in abelian class field theory over the rationals. Composita of quadratic extensions further exemplify abelian groups as direct products. Consider the biquadratic extension \mathbb{Q}(\sqrt{d_1}, \sqrt{d_2})/\mathbb{Q}, where d_1, d_2 are distinct square-free integers not differing by a square factor. This extension has degree 4 and is Galois, with Galois group generated by independent sign changes on each square root, yielding \mathrm{Gal}(\mathbb{Q}(\sqrt{d_1}, \sqrt{d_2})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, the Klein four-group. For example, \mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q} has this group, as the intermediate fields \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{3}), and \mathbb{Q}(\sqrt{6}) correspond to the three subgroups of index 2. More generally, the compositum of k independent quadratics yields (\mathbb{Z}/2\mathbb{Z})^k, a key example of elementary abelian 2-groups in Galois theory.

Non-abelian groups from quartic and symmetric polynomials

Non-abelian Galois groups arise prominently in the study of irreducible quartic polynomials over the rationals \mathbb{Q}, where the possible transitive subgroups of S_4 include the non-abelian groups S_4, A_4, and the dihedral group D_4 of order 8. The Klein four-group V_4 is abelian and thus excluded from this discussion, though it appears as a Galois group for certain quartics like x^4 + 1. To determine the Galois group of an irreducible quartic f(x) \in \mathbb{Q}, one computes its cubic resolvent r(y), which is derived from the action on pairs of roots and has the form y^3 - b y^2 + (a c - 4 d) y - (c^2 - 4 b d + a^2 d) for a monic quartic x^4 + a x^3 + b x^2 + c x + d. If r(y) is irreducible over \mathbb{Q}, the Galois group is either S_4 or A_4; specifically, it is A_4 if the discriminant of f(x) is a square in \mathbb{Q}, and S_4 otherwise. For example, the polynomial x^4 - x - 1 is irreducible over \mathbb{Q} since it is irreducible mod 2, and its resolvent cubic y^3 + 4 y - 1 is irreducible with non-square discriminant -283, yielding Galois group S_4. Similarly, x^4 + 8 x + 12 is irreducible over \mathbb{Q}, has irreducible resolvent y^3 - 48 y - 64, and discriminant $576^2 (a square), so its Galois group is A_4. For the dihedral group D_4, consider x^4 - 3, which is irreducible over \mathbb{Q} (no rational roots and does not factor into quadratics with rational coefficients). Its resolvent cubic y^3 + 12 y factors as y(y^2 + 12) over \mathbb{Q}, and further analysis of the quadratic factor's irreducibility confirms the Galois group as D_4. These examples illustrate how the resolvent cubic distinguishes the non-abelian structures, with S_4 and A_4 corresponding to irreducible resolvents and D_4 to reducible ones with a specific factorization pattern. Beyond quartics, non-abelian Galois groups appear in higher-degree symmetric polynomials, particularly the symmetric groups S_n for n \geq 3 and alternating groups A_n for n \geq 4. For an irreducible polynomial of degree n over \mathbb{Q}, the Galois group embeds as a transitive subgroup of S_n; it is the full S_n if the group contains a transposition (generated, for instance, by the decomposition into irreducible factors modulo some prime), and A_n if it consists of even permutations, verifiable via the discriminant being a square in \mathbb{Q}. For n=3, the alternating group A_3 \cong \mathbb{Z}/3\mathbb{Z} is abelian (cyclic), as seen in the irreducible cubic x^3 - 3x - 1 over \mathbb{Q} (discriminant 81, a square), whose splitting field has degree 3. However, for n=5, A_5 is simple and non-abelian; an example is the irreducible quintic x^5 + 20x + 16 over \mathbb{Q}, with square discriminant and Galois group A_5 confirmed by the irreducibility of its sextic resolvent. The S_5, non-abelian of order 120, occurs for generic irreducible quintics, such as x^5 + x + 1, which is irreducible over \mathbb{Q} (no rational roots, no or cubic factors) and has Galois group S_5 due to containing odd permutations ( not a square) and being transitive. Another example is x^5 - 6x + 3, irreducible by Eisenstein with prime 3, with non-square and a prime yielding a transposition cycle type, ensuring S_5. These cases highlight the prevalence of S_n and A_n for higher symmetric polynomials, underscoring the increasing complexity of non-abelian Galois groups as degree grows. The Q_8 is rare as a Galois group over \mathbb{Q} in characteristic zero and does not arise for irreducible quartics, though it realizes in certain degree-8 extensions.

Infinite Galois groups

In infinite , the \Gal(K/F) of an infinite K/F is equipped with a natural when K is the separable of the base F, rendering it a . Specifically, such a group is the of the finite \Gal(K_i/F) over all finite Galois subextensions K_i/F, where the transition maps are the natural projections. This profinite structure captures the essence of infinite extensions by embedding the finite cases as quotients, ensuring and total disconnectedness in the topological sense. The of a field F, denoted \Gal(\overline{F}/F) where \overline{F} is a separable closure of F, exemplifies an infinite that encodes all finite Galois extensions of F. For instance, \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) is an enormous , known to be non-abelian and to possess uncountably many subgroups of index 2, despite \mathbb{Q} having only countably many quadratic extensions. Its structure remains largely mysterious, with no explicit generators or relations fully known, though it surjects onto every finite group by the for \mathbb{Q}, which is expected but unproven in general. The Krull topology on \Gal(K/F) is defined by taking as a basis the cosets of subgroups \Gal(K/L) for finite Galois extensions L/F within K/F, making open subgroups precisely those corresponding to finite extensions. This topology turns the group into a where continuous homomorphisms correspond to field homomorphisms, and the fundamental theorem of infinite establishes a between intermediate fields and closed subgroups, with fixed fields of closed subgroups yielding the full . In this setting, the fixed field of an open subgroup is a finite extension, highlighting how the topology bridges finite and infinite aspects. Examples of infinite Galois groups abound in specific contexts. For p-adic fields like \mathbb{Q}_p, the absolute Galois group \Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) is profinite, with its maximal unramified quotient isomorphic to the profinite completion \hat{\mathbb{Z}}, reflecting the structure of unramified extensions via powers of the Frobenius. Similarly, over function fields such as \mathbb{F}_q(t), the absolute Galois group is profinite and often free in certain geometric settings, like for curves over finite fields, where it arises as an inverse limit tied to étale fundamental groups. Hilbert's 12th problem seeks to describe the maximal abelian quotient of such groups explicitly using analytic functions, generalizing class field theory from \mathbb{Q} to arbitrary number fields, though full resolutions remain open beyond CM fields.

Computation of Galois Groups

Criteria for determining groups

Determining the Galois group of a separable f(x) \in \mathbb{Q} of n involves identifying its image as a transitive permutation subgroup of S_n. Classical criteria provide tools to detect specific cycle types, subgroup containments, and structural properties without computing the full . These methods rely on arithmetic invariants like factorizations modulo primes and polynomial resolvents, offering partial but decisive information about the group. Dedekind's criterion links the arithmetic of the polynomial modulo primes to the conjugacy classes in the Galois group. For a prime p not dividing the of f(x), the of f(x) modulo p into distinct irreducible factors of degrees d_1, d_2, \dots, d_k corresponds to the cycle type of the Frobenius element \mathrm{Frob}_p in \mathrm{Gal}(K/\mathbb{Q}), where K is the of f(x); specifically, \mathrm{Frob}_p acts as a product of disjoint cycles of lengths d_1, d_2, \dots, d_k on the roots. This allows detection of elements with prescribed cycle structures, such as transpositions (from linear and (n-1)-degree factors) or n-cycles (from irreducibility modulo p), thereby narrowing the possible transitive subgroups of S_n. Irreducibility of f(x) over \mathbb{Q} ensures the Galois group is transitive. By examining multiple such primes, one can often identify the exact group through the presence or absence of certain cycle types. Resolvent polynomials provide another arithmetic tool to distinguish between possible Galois groups, particularly for low degrees. For an irreducible f(x) \in \mathbb{Q}, the associated cubic resolvent g(y) is constructed from the roots of f(x) to capture the action on pairs or triples; its over \mathbb{Q} reveals the structure of the Galois group as a of S_4. If g(y) is irreducible, the Galois group is either S_4 or A_4; reducibility to a linear and factor indicates the D_4, while three linear factors suggest the V_4. This criterion thus separates the possible transitive subgroups without resolving the full extension. The discriminant \Delta of f(x) offers a sign-based criterion for even permutations. For a monic separable f(x) \in \mathbb{Q} of degree n, \Delta is a square in \mathbb{Q} if and only if the Galois group is contained in the A_n. This follows from the fact that the action of \mathrm{Gal}(K/\mathbb{Q}) on the roots induces a homomorphism to S_n, and the kernel of the map corresponds to even permutations; \Delta = \prod_{i < j} (\alpha_i - \alpha_j)^2 (up to sign) changes sign under odd permutations, making it a square precisely when all group elements are even. Combined with other criteria, such as resolvents for quartics, this distinguishes S_4 from A_4. Eisenstein's irreducibility criterion ties local arithmetic to the order of the Galois group but has limitations for full identification. Over \mathbb{Q}, if f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z} satisfies the Eisenstein condition at a prime p (i.e., p \nmid a_{n-1}, \dots, a_1, p \mid a_0, and p^2 \nmid a_0), then f(x) is irreducible, implying the Galois group has order divisible by n and is transitive on the roots. In the p-adic setting over \mathbb{Q}_p, an Eisenstein polynomial generates a totally ramified extension of degree n, where the local Galois group (decomposition group) has order n and is cyclic for tame ramification, providing information on inertia subgroups. However, these local properties constrain only the decomposition and inertia subgroups in the global Galois group, without determining its full structure or distinguishing between non-isomorphic transitive subgroups of order multiple of n.

Algorithms and software tools

One key algorithm for computing Galois groups involves factoring the over finite fields using , which efficiently determines the cycle types of Frobenius elements modulo primes, providing constraints on the possible Galois group as a on the roots. This modular approach, extended by techniques like Hensel lifting for higher precision, allows identification of the group by matching observed cycle structures to transitive subgroups of the . For polynomials defining number fields, p-adic methods offer an alternative by approximating roots in the p-adic numbers to evaluate resolvents or determine orbit structures under the Galois action. Kedlaya's p-adic methods, leveraging , enable efficient computation of Frobenius actions in local settings, aiding the determination of groups and thus the full Galois group for extensions ramified at specific primes. Several software systems implement these algorithms for practical computation. Magma's GaloisGroup function computes the Galois group of irreducible polynomials over up to high degrees using a combination of modular factorizations and resolvent methods, often succeeding for degrees beyond 20. SageMath interfaces with PARI/GP and Magma to compute Galois groups via representations, supporting both absolute and relative extensions. The GAP system excels in identifying groups arising from Galois actions, using databases of transitive groups to match computed cycle indices. In terms of complexity, Galois group computation runs in polynomial time for fixed degree polynomials, relying on the bounded number of transitive subgroups, but becomes exponential in the degree due to the growth in the order of the symmetric group and the need to test multiple candidates.

Applications in number theory and algebra

The inverse Galois problem asks whether every finite group can be realized as the Galois group of some Galois extension of the rational numbers \mathbb{Q}. This problem has been affirmatively solved for the symmetric groups S_n and alternating groups A_n for all n \geq 1, using Hilbert's irreducibility theorem to specialize polynomials over \mathbb{Q}(t) while preserving the Galois group structure. The realization typically involves constructing polynomials whose splitting fields have the desired Galois group, confirming that S_n and A_n arise naturally in number-theoretic contexts over \mathbb{Q}. However, the problem remains open for many other finite groups, such as certain simple non-abelian groups, motivating ongoing research in group theory and arithmetic geometry. In , abelian Galois groups of finite extensions of a number field K are classified via ray class groups, which generalize ideal class groups by incorporating a m that accounts for ramification at specified primes. The Artin reciprocity map provides a isomorphism between the Galois group \mathrm{Gal}(L/K) of an abelian extension L/K and the quotient of the ray class group C_m by the norm subgroup from L, where m is chosen so that the ramified primes divide m. This correspondence, encapsulated in Takagi's existence theorem, ensures that every open of the ray class group corresponds to a unique abelian extension, with the Galois group acting via the Artin map on ideals coprime to m. For the rational field \mathbb{Q}, this recovers Kronecker's theorem on cyclotomic extensions, linking \mathrm{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^\times to the ray class group modulo m. In , étale fundamental groups generalize Galois groups to schemes, capturing the of finite étale covers in a profinite manner. For a C over a \mathbb{F}_q, the étale fundamental group \pi_1^\mathrm{ét}(C_{\overline{\mathbb{F}}_q}) is the profinite completion of the topological when the is zero, but in positive , it incorporates wild ramification and relates to the absolute Galois group of \mathbb{F}_q via a short exact sequence. A key example is the projective line minus three points over \mathbb{F}_q, whose étale fundamental group has finite quotients corresponding to Galois representations from modular forms or hypergeometric functions, aiding the study of point counting and zeta functions via the Frobenius endomorphism. This structure has applications in the Langlands program, where the étale fundamental group encodes arithmetic data of motives associated to the . Galois groups of cyclotomic extensions play a role in through the in subgroups of finite fields constructed using cyclotomic polynomials. These polynomials define irreducible factors over \mathbb{F}_q whose degrees relate to the order of q modulo the cyclotomic conductor, allowing the construction of smooth-order subgroups in \mathbb{F}_q^\times where the DLP underpins secure protocols like Diffie-Hellman. The abelian Galois group (\mathbb{Z}/n\mathbb{Z})^\times of \mathbb{Q}(\zeta_n)/\mathbb{Q} influences the embedding degrees and security parameters in such systems, ensuring efficient computation of logs in small subgroups while maintaining hardness in the full group. This approach enhances the practicality of DLP-based cryptosystems over finite fields by minimizing costs.