Global classfield theory
In this Chapter, k will be an A-field; we use the same notations as in earlier chapters, e.g. k v , r v , q v , k A , etc. We choose an algebraic closure k¯ of k, and, for each place v of k, an algebraic closure K v of k v , containing k¯. We write k sep, k v ,sep for the maximal se¬parable extensions of k in k¯, and of k v in K v , respectively. We write k ab, k v ,ab for the maximal abelian extensions of k in k sep, and of k v in k v ,sep, respectively. One could easily deduce from lemma 1, Chap. XI–3, that k v ,sep is generated over k v by k sep, and therefore K v by k¯, and we shall see in § 9 of this Chapter that k v ,ab is generated over k v by k ab; no use will be made of these facts. We write 𝕲 and 𝔄 = 𝕲/ 𝕲(1) for the Galois groups of k sep and of k ab, respectively, over k; we write 𝕲 v and 𝔄 v = 𝕲 v /𝕲 v (1) for those of k v ,sep and of k v ,ab, respectively, over k v . We write χ v for the re-striction morphism of 𝕲 v into 𝕲, and also, as explained in Chap.XII–1, for that of 𝔄 v into 𝔄. We write X k for the group of characters of 𝕲, or, what amounts to the same, of 𝔄; for each χ ∈ X k , we write χ v =χ°ρ v this is a character of 𝕲 v , or, what amounts to the same, of 𝔄 v .
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