Centralp-Extensions of (p,p,…,p)-Type Galois Groups
نویسندگان
چکیده
منابع مشابه
Galois Groups of Maximal ̂ -extensions
Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th roo...
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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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Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ∼= Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card(R) if L is not formally real. MR Classification: 12E30 Directory: \Jarden\Diary\RealFree 3 April, 2007 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation....
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Let p be a rational prime and S a finite set of rational primes. We are interested in the structure of GS(p), the Galois group of the maximal p-extension of Q unramified outside S (and ∞ if p = 2). In the case that p ∈ S, many GS(p) are known explicitly [12], but in the case that p ∈ S, very little is known. Throughout this report we shall assume that p ∈ S. The author developed methods to comp...
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Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1996
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.0373