Global classfield theory

Abstract

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 k̄ of k, and, for each place v of k, an algebraic closure K _v of k _v , containing k̄ . We write k _sep, k _{v, sep} for the maximal separable extensions of k in k̄, 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 k̄, 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 ρ _v for the restriction 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 = χoρ _v ; this is a character of 𝕲 _v , or, what amounts to the same, of 𝕬 _v .