g] of k' which is obtained as a composite of k' and an abelian extension
of k instead.
Proof: Since E/F is ah elementary abelian extension of degree [p.
Consider an elementary Abelian extension E/F of degree [p.
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.
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.
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 ) the Galois group of an abelian extension
field on a q-dimensional local field K is described by the Milnor K-group [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.