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Kummer theory

Kummer theory is a fundamental framework in algebraic number theory and Galois theory that classifies all finite abelian extensions of a field K of exponent dividing n, assuming K contains the nth roots of unity; these extensions are precisely the fields obtained by adjoining nth roots of elements of K, and they correspond bijectively to subgroups of the multiplicative group K^\times / (K^\times)^n.^[1] In the cohomological formulation, it establishes an isomorphism H^1(G_K, \mu_n) \cong K^\times / (K^\times)^n, where G_K is the absolute Galois group of K and \mu_n is the group of nth roots of unity, enabling the description of such extensions via characters or homomorphisms from G_K to \mathbb{Z}/n\mathbb{Z}.^[2] This theory provides a complete solution to the problem of radical extensions under the given hypothesis, generalizing quadratic field extensions (where n=2) to higher degrees.^[1] Developed by Ernst Kummer (1810–1893) in the mid-19th century, the theory arose from efforts to resolve failures of unique prime factorization in rings of algebraic integers, particularly within cyclotomic fields \mathbb{Q}(\zeta_n), where \zeta_n is a primitive nth root of unity.^[3] Motivated by the pursuit of higher reciprocity laws and proofs of Fermat's Last Theorem (FLT), Kummer introduced the concept of ideal numbers—precursors to Dedekind ideals—to restore unique factorization in these domains, without a full formalization of algebraic integers (a development later completed by Dedekind).^[4] His work built on earlier attempts, such as Euler's flawed analysis of FLT using \mathbb{Z}[\sqrt{-3}], by systematically studying the ring of integers \mathcal{O}_K = \mathbb{Z}[\zeta_n] in cyclotomic fields, proven to hold for all n via theorems on discriminants and induction over prime powers.^[4] A central notion in classical Kummer theory is that of regular primes: an odd prime p is regular if it does not divide the class number of \mathbb{Q}(\zeta_p), ensuring the ring \mathbb{Z}[\zeta_p] behaves well under ideal factorization.^[3] Kummer proved FLT for all regular primes p \geq 5 using this framework, showing that no nontrivial solutions exist to x^p + y^p = z^p by assuming a solution and deriving a contradiction via ideal norms and the non-principal nature of certain ideals raised to the pth power.^[5] He also devised a criterion for irregularity: p divides the numerator of a Bernoulli number B_k for even k from 2 to p-3, allowing identification of irregular primes like 37, 59, and 67 among the first hundred primes.^[3] These results highlighted the role of class groups and units in number fields, with the unit group \mathcal{O}_K^\times finitely generated of rank r_1 + r_2 - 1 (Dirichlet's unit theorem), connecting to cyclotomic units for explicit computations.^[4] Beyond its historical applications to FLT and reciprocity, Kummer theory forms a cornerstone of class field theory, providing the local analogue for abelian extensions and influencing the Kronecker-Weber theorem, which states that every abelian extension of \mathbb{Q} is cyclotomic.^[1] In modern contexts, it extends to function fields, Drinfeld modules, and étale cohomology, with tools like the Kummer pairing—a nondegenerate bilinear map between Galois groups and nth power classes—facilitating computations in Selmer groups and Mordell-Weil ranks of elliptic curves.^[2] The theory's emphasis on Galois cohomology has made it indispensable for studying descent problems and lifting representations, underscoring its enduring impact on arithmetic geometry.^[3]

History and Motivation

Ernst Kummer's Contributions

Ernst Kummer laid the foundations of what is now known as Kummer theory during the 1840s, as part of his pioneering efforts in algebraic number theory focused on cyclotomic fields. His investigations revealed that the ring of integers in these fields often lacks unique factorization, a property essential for classical arithmetic. To address this deficiency, Kummer introduced the concept of ideal numbers around 1846, which allowed him to establish a unique factorization theorem in terms of these ideals, thereby extending arithmetic principles to more complex domains.^[6] The motivation for Kummer's work stemmed directly from the breakdowns in unique factorization observed in the rings of integers of cyclotomic fields, such as the 23rd cyclotomic field, where ordinary integers fail to factor uniquely into irreducibles. These failures disrupted attempts to generalize results from the rational integers, compelling Kummer to devise ideal numbers as abstract entities that behave like primes while preserving factorization properties across the ring. This innovation not only resolved the immediate arithmetic issues but also provided tools for analyzing class groups and units in these fields.^[7]^[8] Early ideas central to Kummer theory appeared in his 1844 paper "De numeris complexis qui radicibus unitatis et numeris integris realibus constant," where he explored the structure of cyclotomic integers and explicitly demonstrated the absence of unique factorization for the case of the 23rd roots of unity. Building on this, Kummer's subsequent publications between 1846 and 1850, including communications to the Berlin Academy, elaborated the ideal number framework, establishing theorems on factorization and equivalence classes of ideals that enable the study of abelian extensions of cyclotomic fields.^[9]

Connection to Fermat's Last Theorem

Kummer developed his theory of ideal numbers, which laid the groundwork for Kummer theory, primarily to address the failure of unique factorization in cyclotomic fields and thereby tackle Fermat's Last Theorem. Specifically, he proved that for an odd prime p, the equation x^p + y^p = z^p has no non-trivial positive integer solutions when p is a regular prime. This result relies on analyzing the arithmetic of the p-th cyclotomic field \mathbb{Q}(\zeta_p), using his theory of ideal numbers to show that any supposed solution would imply the existence of a non-principal ideal whose p-th power is principal, contradicting the regularity condition since all ideals are principal in the class group.^[10]^[11] A prime p is defined as regular if p does not divide the class number h_p of \mathbb{Q}(\zeta_p), meaning the ideal class group has no p-torsion. Kummer established this criterion in 1850, using it to confirm that all odd primes up to 23 are regular, while identifying 37 as the smallest irregular prime, where h_{37} = 37. His work reduced Fermat's Last Theorem to verification for irregular primes, as he demonstrated the theorem holds for all regular primes; although the infinitude of regular primes remains an open conjecture, Kummer's computations covered a significant portion of small primes, leaving only a finite (though growing) list of cases to check manually at the time.^[10]^[6]^[11] A concrete illustration of this connection appears for p = 3, where \mathbb{Q}(\zeta_3) is a Kummer extension of degree 2 over \mathbb{Q} and has class number 1, making 3 regular. Kummer's method decomposes the equation x^3 + y^3 = z^3 into ideals in the ring of integers of \mathbb{Q}(\zeta_3), showing that any non-trivial solution would generate a non-principal ideal, which is impossible due to unique factorization in this field; this aligns with earlier proofs like Euler's but exemplifies the power of Kummer theory for such cases.^[10]^[11]

Prerequisites

Galois Theory Basics for Abelian Extensions

An abelian extension is a Galois extension L/K of fields whose Galois group \mathrm{Gal}(L/K) is an abelian group.^[12] This means that the group operation in \mathrm{Gal}(L/K) is commutative, so for any \sigma, \tau \in \mathrm{Gal}(L/K), \sigma \circ \tau = \tau \circ \sigma. Abelian extensions form an important class in Galois theory because their group structure simplifies the analysis of subextensions and automorphisms.^[13] A special case of abelian extensions are cyclic extensions, where the Galois group \mathrm{Gal}(L/K) is cyclic, meaning it is isomorphic to the additive group \mathbb{Z}/n\mathbb{Z} for some positive integer n.^[13] In such extensions, there exists a generator \sigma \in \mathrm{Gal}(L/K) such that every element is a power of \sigma, and the order of \sigma is n, the degree of the extension [L:K]. Cyclic extensions are fundamental because they capture the simplest non-trivial abelian structures and often arise in the study of roots of unity. The fundamental theorem of Galois theory provides the key framework for understanding abelian extensions. For a finite Galois extension L/K with abelian Galois group G = \mathrm{Gal}(L/K), there is a bijection between the subgroups of G and the intermediate fields M with K \subseteq M \subseteq L: the fixed field of a subgroup H \leq G is the subfield consisting of elements fixed by all elements of H, and the Galois group of L/M is precisely H. Since G is abelian, every subgroup H is normal in G, ensuring that every intermediate extension M/K is itself Galois with abelian Galois group \mathrm{Gal}(M/K) \cong G / H.^[13] Moreover, the degree [L:K] = |G|, and a fundamental property is that the exponent of G—the least common multiple of the orders of its elements—divides |G|, as each element's order divides |G| by Lagrange's theorem. For cyclic groups \mathbb{Z}/n\mathbb{Z}, the exponent equals n, matching the degree directly.^[14] For infinite abelian extensions, infinite Galois theory extends these ideas by viewing the Galois group as a profinite group, an inverse limit of finite groups corresponding to finite subextensions.^[15] This topological structure preserves the correspondence between closed normal subgroups and fixed fields, though the full details lie beyond finite cases.

Roots of Unity and Cyclotomic Fields

In a field K whose characteristic does not divide n, the group of nth roots of unity, denoted \mu_n, consists of all elements \zeta in the algebraic closure \overline{K} of K satisfying \zeta^n = 1.^[16] These roots form a cyclic group of order n under multiplication, generated by any primitive nth root of unity, which is an element of order exactly n.^[17] The nth cyclotomic field is the extension \mathbb{Q}(\zeta_n) of the rationals \mathbb{Q} obtained by adjoining a primitive nth root of unity \zeta_n.^[16] This extension is Galois over \mathbb{Q}, with Galois group isomorphic to (\mathbb{Z}/n\mathbb{Z})^\times, the multiplicative group of integers modulo n.^[17] The degree of the 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 coprime to n.^[17] The minimal polynomial of \zeta_n over \mathbb{Q} is the nth cyclotomic polynomial, defined as

\Phi_n(x) = \prod (x - \zeta),

where the product runs over all primitive nth roots of unity \zeta.^[16] This polynomial is monic, irreducible over \mathbb{Q}, and has degree \phi(n).^[17] A key property relevant to Kummer theory is that if a base field K already contains the full group \mu_n, then adjoining an nth root of an element in K that is not already an nth power produces a cyclic Galois extension of K of degree dividing n.^[5]

Kummer Extensions

Definition and Basic Properties

A Kummer extension is a field extension L/K obtained by adjoining an nth root of an element a \in K, that is, L = K(\alpha) where \alpha^n = a, under the assumption that the field K contains all nth roots of unity \mu_n \subseteq K and the characteristic of K does not divide n.^[1]^[18] In this setting, the extension has degree dividing n, specifically [L : K] = m where m divides n and m is the smallest positive integer such that a is an mth power in K.^[1] If the minimal polynomial of \alpha over K is separable, then L/K is a Galois extension.^[18] The Galois group \mathrm{Gal}(L/K) of a Kummer extension L = K(\alpha) with [\ L : K\ ] = m is cyclic of order m, generated by the automorphism \sigma defined by \sigma(\alpha) = \zeta \alpha, where \zeta is a primitive mth root of unity in \mu_n \subseteq K.^[1]^[18] This action extends uniquely to all of L since \{1, \alpha, \dots, \alpha^{m-1}\} forms a basis over K, and the group structure reflects the abelian nature of the extension, with every element satisfying \sigma^m = \mathrm{id}.^[1] More generally, a finite abelian extension L/K of exponent dividing n (meaning every element of \mathrm{Gal}(L/K) has order dividing n) with \mu_n \subseteq K is a Kummer extension, generated by adjoining all nth roots of elements from a finite subgroup \Delta of K^\times / (K^\times)^n, so L = K(\Delta^{1/n}).^[1]^[19] In this case, the degree [L : K] = |\Delta|, and the extension is Galois with abelian Galois group of exponent dividing n.^[1] The subgroup \Delta can be taken as \Delta = \{ b (K^\times)^n \mid b \in K^\times \cap (L^\times)^n \}, which captures the elements of K^\times whose nth powers generate the extension.^[18] Kummer extensions are solvable, as their Galois groups are abelian (hence solvable), and any extension solvable by radicals can be decomposed into a tower of such Kummer steps when the base field contains the necessary roots of unity.^[1] This solvability property underscores their role in understanding radical extensions within the broader framework of Galois theory.^[19]

Examples of Kummer Extensions

Kummer extensions arise when adjoining nth roots to a base field K that already contains the nth roots of unity \mu_n. The simplest case occurs for n=2, where \mu_2 = \{\pm 1\} \subseteq K, such as any field of characteristic not 2, including the rationals \mathbb{Q}. In this setting, quadratic extensions of the form K(\sqrt{a})/K for a \in K^\times \setminus (K^\times)^2 are Kummer extensions with Galois group isomorphic to \mathbb{Z}/2\mathbb{Z}. For example, over \mathbb{Q}, the extension \mathbb{Q}(\sqrt{2})/\mathbb{Q} is a quadratic Kummer extension, generated by adjoining the square root of 2, which is not a square in \mathbb{Q}.^[20] For higher degrees, the presence of \mu_n in K is crucial. Over \mathbb{Q}, there are no cyclic extensions of degree 3 that are Kummer extensions because \mu_3 \not\subseteq \mathbb{Q}; the primitive cube roots of unity lie in the quadratic extension \mathbb{Q}(\zeta_3), where \zeta_3 = e^{2\pi i / 3}. However, over K = \mathbb{Q}(\zeta_3), which has degree 2 over \mathbb{Q} and contains \mu_3, adjoining a cube root of an element not a cube yields a Kummer extension. A concrete example is

L = \mathbb{Q}(\zeta_3, \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})/\mathbb{Q}(\zeta_3)

, which has degree 3 and Galois group cyclic of order 3, since 2 is cube-free and not a cube in \mathbb{Q}(\zeta_3).^[21]^[22] More generally, Kummer extensions can involve adjoining multiple nth roots simultaneously. If \Delta = \{a_1, \dots, a_k\} \subseteq K^\times generates a subgroup of K^\times / (K^\times)^n of rank r, then the extension L = K(\sqrt{a_1}, \dots, \sqrt{a_k})/K is a Kummer extension of degree n^r with abelian Galois group of exponent dividing n. For instance, over \mathbb{Q}(\zeta_5), adjoining fifth roots of two independent elements (modulo fifth powers) produces a degree-25 extension that is the compositum of two cyclic quintic Kummer extensions.^[20] Cyclotomic extensions themselves provide historical examples of Kummer extensions when the base field already contains appropriate roots of unity. For an odd prime p, the extension \mathbb{Q}(\zeta_{p^2})/\mathbb{Q}(\zeta_p) is a Kummer extension of degree p, obtained by adjoining a pth root of \zeta_p, with Galois group \mathbb{Z}/p\mathbb{Z}. This illustrates how Kummer theory applies to layers of cyclotomic towers, connecting to broader developments in class field theory.^[2]

The Kummer Correspondence

The Kummer Map

In Kummer theory, the Kummer map provides a fundamental connection between the multiplicative structure of the base field and the Galois group of a Kummer extension. Consider a Galois extension L/K of exponent n, where the field K contains the group \mu_n of all nth roots of unity and \operatorname{char}(K) does not divide n. The Kummer map \delta: K^\times / (K^\times)^n \to \operatorname{Hom}(\operatorname{Gal}(L/K), \mu_n) is defined as follows: for a class represented by a \in K^\times, choose \alpha \in L such that \alpha^n = a; then \delta(a)(\sigma) = \sigma(\alpha)/\alpha for each \sigma \in \operatorname{Gal}(L/K).^[1] This definition is independent of the choice of \alpha, since any other \alpha' = \alpha \zeta for \zeta \in \mu_n yields \sigma(\alpha')/\alpha' = \sigma(\alpha)/\alpha, as elements of \mu_n are fixed by \operatorname{Gal}(L/K).^[1] The map \delta is a group homomorphism, and its kernel is precisely (K^\times)^n.^[2] When L/K is cyclic and generated by a single nth root, that is, L = K(\sqrt{a}) for some a \in K^\times, the map \delta is injective.^[23] In the broader setting of the Kummer correspondence, where L = K(\sqrt{a_1}, \dots, \sqrt{a_d}) for a finite set \{a_i\} generating the extension, \delta is surjective onto the continuous homomorphisms factoring through \operatorname{Gal}(L/K).^[1] From a cohomological perspective, the Kummer map extends to \delta: L^\times / (L^\times)^n \to H^1(\operatorname{Gal}(L/K), \mu_n), defined via the coboundary in the long exact sequence arising from the short exact sequence $1 \to \mu_n \to L^\times \xrightarrow{(\cdot)^n} L^\times \to 1.^[2] Here, \mu_n is a trivial \operatorname{Gal}(L/K)-module, so H^1(\operatorname{Gal}(L/K), \mu_n) \cong \operatorname{Hom}(\operatorname{Gal}(L/K), \mu_n). The triviality of H^1(\operatorname{Gal}(L/K), L^\times) follows from Hilbert's Theorem 90, ensuring the connecting homomorphism \delta is well-defined and captures the relevant cohomology classes.^[2] Specifically, for a cocycle f: \operatorname{Gal}(L/K) \to \mu_n given by f(\sigma) = \sigma(\beta)/\beta with \beta \in L^\times, the cohomology class $$ is the image \delta(\beta^n).^[23]

Isomorphism with Homomorphisms

The Kummer correspondence theorem establishes an anti-isomorphism between the lattice of subgroups of K^\times / (K^\times)^n and the lattice of closed subgroups of \Hom_{\cont}(\Gal(\bar{K}/K), \mu_n), where K is a field containing the group of nth roots of unity \mu_n, thereby classifying all finite abelian extensions of K of exponent dividing n. Specifically, each finite subgroup \Delta \leq K^\times / (K^\times)^n corresponds to the Kummer extension L_\Delta = K(\sqrt{a} \mid \in \Delta), with \Gal(L_\Delta / K) \cong \Hom(\Delta, \mu_n), and the anti-isomorphism arises via annihilators of the image under the Kummer map \delta: K^\times / (K^\times)^n \to \Hom_{\cont}(\Gal(\bar{K}/K), \mu_n).^[1] For a fixed Kummer extension L/K generated by nth roots of elements in K, the subgroup \Delta = K^\times \cap (L^\times)^n / (K^\times)^n is canonically isomorphic to \Hom(\Gal(L/K), \mu_n), with the isomorphism induced by the restriction of the Kummer map from the previous section. The proof that this map \delta: \Delta \to \Hom(\Gal(L/K), \mu_n) is an isomorphism relies on classical Galois theory. First, \delta is a group homomorphism because the action of \Gal(L/K) on roots is multiplicative: for classes , \in \Delta, \delta([ab])(\sigma) = \sigma((ab)^{1/n}) / (ab)^{1/n} = \delta()(\sigma) \cdot \delta()(\sigma) for all \sigma \in \Gal(L/K). Injectivity follows from the faithful action of \Gal(L/K) on the roots: if \delta() = 1, then \sigma(a^{1/n}) = a^{1/n} for all \sigma, implying a^{1/n} \in K and thus = 1 in \Delta. Surjectivity is shown by constructing, for any \chi \in \Hom(\Gal(L/K), \mu_n), an element a \in K^\times such that the fixed field of \ker \chi is K(a^{1/n}), using the fact that characters determine cyclic subextensions corresponding to radicals.^[24] When n = p is prime (and \operatorname{char} K \neq p), both \Delta and \Hom(\Gal(L/K), \mu_p) are vector spaces over \mathbb{F}_p, and the isomorphism preserves dimension: \dim_{\mathbb{F}_p} \Delta = \dim_{\mathbb{F}_p} \Hom(\Gal(L/K), \mu_p), which equals the minimal number of generators of \Gal(L/K) as a \mathbb{Z}/p\mathbb{Z}-module. This matching of dimensions confirms the linear algebra structure underlying the group-theoretic correspondence. In general, for an abelian extension L/K of exponent dividing n, \Gal(L/K) \cong (\mathbb{Z}/n\mathbb{Z})^r where r = \dim_{\mathbb{F}} \Delta and \mathbb{F} = \mathbb{Z}/n\mathbb{Z} viewed as a field when n is prime (or more generally, as the ring \mathbb{Z}/n\mathbb{Z}). If \{ [a_1], \dots, [a_r] \} forms a basis for \Delta over \mathbb{F}, the dual basis \{ \chi_1, \dots, \chi_r \} for \Hom(\Gal(L/K), \mu_n) satisfies \chi_i(\sigma_j) = \zeta for a primitive nth root of unity \zeta, where \sigma_j generates the cyclic component corresponding to a_j; this pairing explicitly links the radicals to the character basis.^[24] The finite case extends to infinite abelian extensions via profinite completion: the isomorphism passes to the direct limit over finite subextensions, yielding a correspondence between profinite \mathbb{Z}_n-modules and suitable completions of subgroups of K^\times / (K^\times)^{n^\infty}.

Recovering nth Roots

From Primitive Elements

In Kummer theory, for a prime p such that the field K contains the group \mu_p of pth roots of unity, a cyclic extension L/K of degree p admits a primitive element \beta \in L whose minimal polynomial over K has degree p. To recover an element \alpha \in L satisfying \alpha^p \in K from this primitive element, consider the Galois group \mathrm{Gal}(L/K) \cong \mathbb{Z}/p\mathbb{Z} generated by an automorphism \sigma. Define

\alpha = \sum_{k=0}^{p-1} \zeta_p^{-k} \sigma^k(\beta),

where \zeta_p is a primitive pth root of unity in K. This construction yields \alpha^p \in K, and moreover, \sigma(\alpha) = \zeta_p \alpha, confirming that \alpha behaves as a pth root scaled by the action of the generator \sigma.^[25] The formula isolates the radical by leveraging the cyclic Galois action on \beta. Applying \sigma to \alpha gives

\sigma(\alpha) = \sum_{k=0}^{p-1} \zeta_p^{-k} \sigma^{k+1}(\beta) = \sum_{k=1}^{p} \zeta_p^{-(k-1)} \sigma^k(\beta) = \zeta_p \sum_{k=0}^{p-1} \zeta_p^{-k} \sigma^k(\beta) = \zeta_p \alpha,

using the cyclicity of \sigma^p = \mathrm{id} and the relation \zeta_p^p = 1. To verify \alpha^p \in K, note that the powers \sigma^j(\alpha) = \zeta_p^j \alpha for j = 0, \dots, p-1 span the conjugates, and the elementary symmetric functions in these are fixed by \sigma, hence lie in K. The key orthogonality property of roots of unity ensures the sum is nonzero and generates the desired radical: for any character \chi of \mathbb{Z}/p\mathbb{Z} given by \chi(\sigma^k) = \zeta_p^{m k}, the sum \sum_{k=0}^{p-1} \chi(k) = p if \chi is trivial and 0 otherwise, projecting onto the eigenspace for the trivial representation.^[25] This method assumes \mu_p \subseteq K, as the roots of unity are required to define the weighted sum and ensure the extension is radical. The element \alpha generates L over K(\alpha^p), establishing L = K(\alpha) as a Kummer extension. For the case p=2, with \zeta_2 = -1, the formula simplifies to \alpha = \beta - \sigma(\beta), and \alpha^2 = \beta^2 + \sigma(\beta)^2 - 2 \beta \sigma(\beta) \in K since \sigma fixes elements of K and interchanges the quadratic conjugates. This recovers the square root up to units in K, adjusted by the characteristic not dividing 2 to avoid degeneracy.^[25]

For Composite Exponents

When the exponent n is composite and square-free, i.e., n = \prod p_i for distinct primes p_i, the recovery of nth roots in a Kummer extension L/K of degree n proceeds by decomposing L into a chain of prime-degree subextensions corresponding to the prime factors of n. The Galois group \mathrm{Gal}(L/K) \cong \mathbb{Z}/n\mathbb{Z} admits a composition series with factors \mathbb{Z}/p_i\mathbb{Z}, allowing stepwise construction of the radicals.^[26] For each prime p_i, consider the Sylow p_i-subgroup of \mathrm{Gal}(L/K), which has order p_i. The fixed field M_i of this subgroup satisfies [M_i : K] = n/p_i, and L/M_i is a cyclic Kummer extension of prime degree p_i. Applying the recovery method for prime exponents to this subextension yields an element a_i \in M_i such that L = M_i(a_i^{1/p_i}).^[2] To combine these into a single nth root over K, form A = \prod_i a_i^{n/p_i} \in K. Then L = K(A^{1/n}), as the exponents n/p_i (coprime to p_i) ensure that the Galois action on A^{1/n} generates the full cyclic group of order n. This requires \mu_n \subseteq K, the group of nth roots of unity. Given a primitive element \beta \in L, the elements a_i can be obtained via traces from \beta to the fixed fields M_i.^[23] For the case n = pq with distinct primes p, q, first recover the pth root over the fixed field M of the q-Sylow subgroup ([M : K] = q), so L = M(b^{1/p}) for some b \in M. Next, recover the qth root over the fixed field N of the p-Sylow subgroup ([N : K] = p), so L = N(c^{1/q}) for some c \in N. Set A = b^q c^p \in K, yielding L = K(A^{1/(pq)}).^[26] This construction assumes n square-free for simplicity; when n involves higher prime powers, recovery necessitates iterated extensions beyond a single radical.^[2]

Applications

To Elliptic Curves

Kummer theory extends naturally to elliptic curves through the lens of Galois cohomology, providing a powerful tool for studying the arithmetic of rational points on these curves. For an elliptic curve E defined over a number field K with \mu_n \subseteq K (ensuring the n-torsion submodule E is a constant Galois module), consider the short exact sequence of étale sheaves on \Spec K: $0 \to E \to E \xrightarrow{n} E \to 0, where n denotes the multiplication-by-n map. Taking Galois cohomology with respect to the absolute Galois group G_K = \Gal(\bar{K}/K) yields the long exact sequence, from which the Kummer sequence extracts the exact sequence $0 \to E(K)/n E(K) \to H^1(K, E) \to H^1(K, E) \to 0. This sequence is exact when the characteristic of K does not divide n, and it underpins descent methods for elliptic curves.^[27] The connecting homomorphism in this sequence, known as the Kummer map \delta: E(K)/n E(K) \to H^1(K, E), embeds the quotient of rational points modulo n-multiples into the first cohomology group. Elements of H^1(K, E) classify principal homogeneous spaces (torsors) under E, or equivalently, certain n-isogenies from elliptic curves to E, up to isomorphism. The image of the Kummer map corresponds precisely to those torsors that arise as n-covers of E by translates of itself, linking the geometry of E directly to its rational points. This setup assumes \mu_n \subseteq K for simplicity, as the cyclotomic action on E simplifies the Galois module structure and facilitates explicit computations.^[27]^[28] A key application lies in measuring the n-Selmer group \Sel_n(E/K), defined as the subgroup of H^1(K, E) consisting of classes that map to zero in H^1(K_v, E) for all places v of K (i.e., locally trivial classes). The Kummer sequence restricts to the exact sequence $0 \to E(K)/n E(K) \to \Sel_n(E/K) \to \Sha(E/K) \to 0, where \Sha(E/K) is the Tate–Shafarevich group. By the Mordell–Weil theorem, which states that E(K) \cong \mathbb{Z}^r \oplus T for finite torsion T, the order of E(K)/n E(K) is n^r times a bounded factor depending on T. Thus, the finiteness of \Sel_n(E/K) (often computable via descent) provides an upper bound on the rank r of E(K), and equality holds if \Sha(E/K) = 0. This framework bounds the rank and aids in finding generators for E(K), connecting Kummer theory to the distribution of rational points on E.^[27]^[28] For the specific case n=2, the Kummer sequence facilitates 2-descent, a classical method to compute the 2-Selmer group and determine the 2-primary part of the rank over \mathbb{Q}. Assuming full 2-torsion over \mathbb{Q}, the map

\delta: E(\mathbb{Q})/2 E(\mathbb{Q}) \to H^1(\mathbb{Q}, E{{grok:render&&&type=render_inline_citation&&&citation_id=2&&&citation_type=wikipedia}}) \cong \mathbb{Q}^\times / (\mathbb{Q}^\times)^2 \oplus \mathbb{Q}^\times / (\mathbb{Q}^\times)^2

(via the Weil pairing) reduces to solving quartic equations representing homogeneous spaces of the form y^2 = x(x^2 + a x + b). Solvable classes yield the 2-rank, with examples like the curve y^2 = x^3 + x illustrating rank computation via explicit 2-descent, often revealing rank 0 or 1 over \mathbb{Q}. This links directly to rational points, as nontrivial elements in the Selmer group may obstruct or generate points on E.^[29]^[30]

In Class Field Theory

Kummer theory plays a central role in class field theory by providing an explicit realization of the Artin reciprocity map for cyclic extensions of exponent n. In this context, for a field K containing the nth roots of unity \mu_n, the theory establishes a bijection between finite abelian extensions of K of exponent n and subgroups of K^\times / (K^\times)^n, where the Galois group of such an extension L/K is isomorphic to the dual of the corresponding quotient via the Kummer pairing. This correspondence realizes the local or global Artin map as an isomorphism from the idele class group (or ray class group) onto the Galois group of the maximal abelian extension, with norm groups determining the ramification.^[31] For local fields K containing \mu_n, Kummer theory identifies the maximal abelian extension of exponent n as K(\Delta^{1/n}), where \Delta generates K^\times / (K^\times)^n N_{L/K}(L^\times) for suitable L. This construction matches the predictions of local class field theory, where the Artin map \phi: K^\times \to \mathrm{Gal}(L/K)^\mathrm{ab} is surjective with kernel the norm group N_{L/K}(L^\times), ensuring that cyclic extensions correspond precisely to quotients of K^\times by norms and powers. In particular, for non-archimedean local fields, this yields totally ramified extensions like K(\pi^{1/n}) for a uniformizer \pi, with degree dividing n.^[31] In the global setting, over \mathbb{Q}(\zeta_n), unramified abelian extensions correspond to quotients of the class group, and Kummer theory provides explicit radical generators for the ray class fields, linking them to ideals in the ray class group modulo n. The Artin reciprocity map factors through the idele class group C_K, inducing isomorphisms for these Kummer extensions. This framework extends to ray class groups C_m, where subgroups yield abelian extensions with controlled ramification outside primes dividing m.^[31] A key consequence is the Kronecker-Weber theorem, which asserts that every abelian extension of \mathbb{Q} is contained in a cyclotomic field \mathbb{Q}(\zeta_m) for some m; since \mathbb{Q}(\zeta_m) contains \mu_m, all such extensions are Kummer extensions over \mathbb{Q}(\zeta_m), with Galois groups realized via units and class groups. For an explicit example, the ray class field of \mathbb{Q} modulo n is precisely \mathbb{Q}(\zeta_n), whose Galois group over \mathbb{Q} is isomorphic to (\mathbb{Z}/n\mathbb{Z})^\times, generated by roots of unity and confirming the reciprocity law through Kummer radicals.^[32]

Generalizations

Cohomological Framework

The cohomological framework reformulates Kummer theory within the broader context of Galois cohomology, providing a unified language for classifying abelian extensions via cohomology groups. For a Galois extension L/K with Galois group G = \mathrm{Gal}(L/K) and a discrete [G-module](/page/G-module) M, the first cohomology group H^1(G, M) consists of continuous crossed homomorphisms from G to M modulo principal ones, and it classifies G-torsors under M or, in certain cases, equivalence classes of extensions of K by cyclic groups acting on M.^[33] This setup generalizes classical Galois theory by associating cohomological invariants to field extensions, where elements of H^1(G, M) correspond to cocycles representing the action in the extension.^[34] Kummer theory emerges as a special case when M = \mu_n, the group of nth roots of unity in an algebraic closure \bar{K} of K, assuming \mu_n \subset K. Consider the short exact sequence of G-modules

$0 \to \mu_n \to \bar{K}^\times \xrightarrow{x \mapsto x^n} \bar{K}^\times \to 0,

where G = \mathrm{Gal}(\bar{K}/K). The long exact cohomology sequence yields a connecting homomorphism \delta: K^\times / (K^\times)^n \to H^1(G, \mu_n), which is an isomorphism.^[2] This identifies cyclic extensions of exponent dividing n with elements of K^\times / (K^\times)^n, where the image of \alpha \in K^\times under \delta corresponds to the Kummer extension K(\sqrt{\alpha})/K. A key ingredient is Hilbert's Theorem 90, which states that H^1(G, \bar{K}^\times) = 0 (the trivial group), ensuring the exactness at that position and thus the isomorphism for the Kummer map.^[34] For the infinite-dimensional case, encompassing all finite abelian extensions under the assumption that K contains all roots of unity, one considers the direct limit over n of the above sequences, leading to the module \mathbb{Q}/\mathbb{Z}(1) = \varinjlim \mu_n. The infinite Kummer map is then \delta: K^\times \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z} \to H^1(G, \mathbb{Q}/\mathbb{Z}(1)), which is an isomorphism. With the trivial action on \mathbb{Q}/\mathbb{Z}(1), this yields H^1(G, \mathbb{Q}/\mathbb{Z}(1)) \cong \mathrm{Hom}_{\mathrm{cont}}(G, \mathbb{Q}/\mathbb{Z}), classifying all finite abelian extensions of K.^[35] In this setting, the \mathbb{Z}-rank (or dimension over \mathbb{Q}) of H^1(G, \mu_n) equals the minimal number of independent nth radicals required to generate the corresponding maximal abelian extension of exponent n.^[2]

Artin-Schreier Analogue

In fields of characteristic p > 0, the Artin-Schreier theory provides an additive analogue to Kummer theory, describing cyclic Galois extensions of degree p. An Artin-Schreier extension is a field extension L = K(\beta) where \beta satisfies the equation \beta^p - \beta = a for some a \in K not in the image of the map x \mapsto x^p - x on K, ensuring irreducibility and yielding a Galois group isomorphic to \mathbb{Z}/p\mathbb{Z}.^[34] The Galois action is generated by \sigma(\beta) = \beta + 1, reflecting the additive structure of the prime field \mathbb{F}_p.^[36] The analogy to Kummer theory lies in the correspondence between such extensions and homomorphisms from the Galois group to \mathbb{F}_p. Define the Artin-Schreier map \delta: K / (K^p - K) \to \Hom(\Gal(L/K), \mathbb{F}_p) by \delta(a)(\sigma) = \sigma(\beta) - \beta, which captures the additive deviation under Galois action, mirroring the multiplicative Kummer map \chi: K^\times / (K^\times)^p \to \Hom(\Gal(L/K), \mu_p) but replacing roots of unity with the additive group \mathbb{F}_p.^[34] This setup classifies all cyclic extensions of degree p in characteristic p.^[36] For abelian extensions of exponent p, or more generally of p-power degree with elementary abelian Galois groups, the theory extends via Artin-Schreier-Witt extensions, which parameterize them using Witt vectors over \mathbb{F}_p. Every such extension arises as a compositum of Artin-Schreier extensions, with the Galois group corresponding to subgroups of the Witt vector module modulo the image of the Verschiebung map.^[34] Unlike Kummer theory, there is no obstruction from roots of unity, as the p-torsion in the multiplicative group is trivial in characteristic p, facilitating the complete description of these extensions and their role in the class field theory of positive characteristic fields.^[34] A representative example occurs over the function field K = \mathbb{F}_p(t), where adjoining a root \beta of x^p - x - 1/t = 0 yields a cyclic extension of degree p, as $1/t lies outside the image of the Artin-Schreier map on K.^[34]

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