Shafarevich–Weil theorem

In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by Shafarevich (1946) for local fields and by Weil (1951) for global fields.

Statement[edit]

Suppose that F is a global field, K is a normal extension of F, and L is an abelian extension 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 group H²(Gal(K/F), Gal(L/K)). On the other hand, class field theory 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 (Artin & Tate 2009, p.246).

Shafarevich stated his theorem for local fields in terms of division algebras rather than the fundamental class (Weil 1967). In this case, with L the maximal abelian extension of K, the extension Gal(L/F) corresponds under the reciprocity map to the normalizer 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) and under this correspondence the division algebra with Hasse invariant 1/[K:F] corresponds to the fundamental class.

References[edit]

Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335
Shafarevich, I. R. (1946), "On Galois groups of p-adic fields.", C. R. (Doklady) Acad. Sci. URSS, New Series, 53: 15–16, MR 0018170 Reprinted in his collected works, pages 4–5
Weil, André (1951), "Sur la theorie du corps de classes", Journal of the Mathematical Society of Japan, 3: 1–35, doi:10.2969/jmsj/00310001, ISSN 0025-5645, MR 0044569, reprinted in volume I of his collected papers, ISBN 0-387-90330-5
Weil, André (1967), "Appendix III:Shafarevitch's theorem", Basic number theory., Die Grundlehren der mathematischen Wissenschaften, vol. 144, Springer-Verlag New York, Inc., New York, pp. 301–307, ISBN 3-540-58655-5, MR 0234930