Galois groups of unramified solvable extensions
نویسندگان
چکیده
منابع مشابه
CONSTRUCTION OF MAXIMAL UNRAMIFIED p-EXTENSIONS WITH PRESCRIBED GALOIS GROUPS
For any number field F (not necessary of finite degree) and prime number p, let Lp(F ) denote the maximal unramified p-extension over F , and put G̃F (p) = Gal(Lp(F )/F ). Though the structure of G̃F (p) has been one of the most fascinating theme of number theory, our knowledge on it is not enough even at present: It had been a cerebrated open problem for a long time whether G̃F (p) can be infinit...
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Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.
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ژورنال
عنوان ژورنال: Tohoku Mathematical Journal
سال: 1982
ISSN: 0040-8735
DOI: 10.2748/tmj/1178229257