algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, the Shafarevich–Weil theorem relates the fundamental class of a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...

global Global may refer to: General *Globe, a spherical model of celestial bodies *Earth, the third planet from the Sun Entertainment * ''Global'' (Paul van Dyk album), 2003 * ''Global'' (Bunji Garlin album), 2007 * ''Global'' (Humanoid album), 198 ...

fields to an extension of

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s. It was introduced by for local fields and by for global fields.

Statement

Suppose that ''F'' is a global field, ''K'' is a

normal extension In abstract algebra, a normal extension is an Algebraic extension, algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' that has a zero of a function, root in ''L'' splits into linear factors over ''L''. This is ...

of ''F'', and ''L'' is an

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...

of ''K''. Then the Galois group Gal(''L''/''F'') is an extension of the group Gal(''K''/''F'') by the abelian group Gal(''L''/''K''), and this extension corresponds to an element of the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

group H²(Gal(''K''/''F''), Gal(''L''/''K'')). On the other hand,

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

gives a fundamental class in H²(Gal(''K''/''F''),''I''_''K'') and a reciprocity law map from ''I''_''K'' to Gal(''L''/''K''). The Shafarevich–Weil theorem states that the class of the extension Gal(''L''/''F'') is the image of the fundamental class under the homomorphism of cohomology groups induced by the reciprocity law map . Shafarevich stated his theorem for local fields in terms of

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

s rather than the fundamental class . In this case, with ''L'' the maximal abelian extension of ''K'', the extension Gal(''L''/''F'') corresponds under the reciprocity map to the

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

of ''K'' in a division algebra of degree 'K'':''F''over ''F'', and Shafarevich's theorem states that the Hasse invariant of this division algebra is 1/ 'K'':''F'' The relation to the previous version of the theorem is that division algebras correspond to elements of a second cohomology group (the

Brauer group In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...

) and under this correspondence the division algebra with Hasse invariant 1/ 'K'':''F''corresponds to the fundamental class.

References

* * Reprinted in his collected works, pages 4–5 * , reprinted in volume I of his collected papers, * {{DEFAULTSORT:Shafarevich-Weil theorem Theorems in algebraic number theory Class field theory