Abelian extension


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Abelian extension

[ə′bēl·yən ik′sten·chən]
(mathematics)
A Galois extension whose Galois group is Abelian.
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Proof: Since E/F is ah elementary abelian extension of degree [p.sup.n], then
Let N/K be a finite abelian extension of number fields of group G.
Let N/K be an abelian extension of number fields of group G, and let H be a subgroup of G.
We denote by [Z.sub.l] the ring of l-adic integers, and by [B.sub.[infinity]] the unique abelian extension of Q in C whose Galois group over Q is topologically isomorphic to the additive group of Z/.
Next let F be a finite abelian extension over Q in C.
Let L/k be an abelian extension that contains K and splits A.
But [square root of [DELTA]] [member of] [Q.sup.ab] = [Q.sup.cyc], where [Q.sup.ab] is the maximal abelian extension of Q.
Then by Proposition 1.6, [L.sub.0][K.sub.[infinity]] is an abelian extension of M.
For example, one may cite the study of the maximal unramified extension of a local field or the maximal abelian extension of a global field.
Finally in the last section we reach a factorization formula which is an analog of the decomposition of the Dedekind zeta function of an abelian extension into Hecke L-functions.
Kato ([5,6] and [7]) the Galois group of an abelian extension field on a q-dimensional local field K is described by the Milnor K-group [K.sup.M.sub.q] (K) for q [greater than or equal to] [1.sup.*1] The information on the ramification is related to the natural filtration [U.sup.m] [K.sub.q] = [U.sup.m] [K.sup.M.sub.q] (K) which is by definition the subgroup generated by {1 + [m.sup.m.sub.K],[K.sup.x], ..., [K.sup.x]},where [m.sub.K] is the maximal ideal of the ring of integers [O.sub.k].
Among the topics are quadratic points of classical modular curves, p-adic point counting on singular super-elliptic curves, a vanishing criterion for Dirichlet series with periodic coefficients, the Sato-Tate conjecture for a Picard curve with a complex multiplication, arithmetic twists with abelian extensions, and transcendental numbers with special values of Dirichlet series.