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Results tagged with class-field-theory
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user 5015
13
votes
Accepted
Finite Galois module whose Ш¹ is nonzero?
Wang's conterexample to Grunwald's theorem: $K=\mathbb{Q}(\sqrt{7})$ and $M=\mu_8$. Then $H^1(K,M) \cong K^\times/(K^\times)^8$. Now $16$ is not an $8$-th power in this field but locally an $8$-th pow …
- 8,945
8
votes
Accepted
Capitulation in cyclotomic extensions
Assume $p$ is an irregular prime for which
Vandiver's conjecture holds, e.g. $p<12'000'000$. This conjecture asserts that $p$ does not divide the $+$-part of the class group.
Then there is no capitu …
- 8,945
2
votes
class group size of cyclotomic field subextension
The $p$-primary part of the class group of $\mathbb{Q}_1$, and in fact all $\mathbb{Q}_n$ in the cyclotomic tower of $\mathbb{Q}$, is trivial for all $p$. This is contained in Proposition 13.22 of Was …
- 8,945