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On the Galois cohomology of ideal class groups

✍ Scribed by David Burns; Soogil Seo

Publisher
Springer
Year
2007
Tongue
English
Weight
138 KB
Volume
89
Category
Article
ISSN
0003-889X

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It is well known that the Galois group of an extension L/F puts constraints on the structure of the relative ideal class group Cl(L/F ). Explicit results, however, hardly ever go beyond the semisimple abelian case, where L/F is abelian (in general cyclic) and where (L : F ) and #Cl(L/F ) are coprime

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Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group G F (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H\*(G F , F 2 ) contains the mod

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We prove that two arithmetically significant extensions of a field F coincide if w x and only if the Witt ring WF is a group ring β€«ήšβ€¬rn G . Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's Theorem 90 and show that an identity li