Journal de Th´
eorie des Nombres
de Bordeaux
18
(2006), 627–652
Constructing class fields over local fields
par
Sebastian Pauli
Dedicated to Michael Pohst on his 60th Birthday
R´esum´e. Soit K un corps p-adique. Nous donnons une carac-t´erisation explicite des extensions ab´eliennes de K de degr´epen reliant les coefficients des polynˆomes engendrant les extensions L/K de degr´e p aux exposants des g´en´erateurs du groupe des normes NL/K(L∗). Ceci est appliqu´e `a un algorithme de
con-struction des corps de classes de degr´epm, ce qui conduit `a un algorithme de calcul des corps de classes en g´en´eral.
Abstract. LetK be a p-adic field. We give an explicit charac-terization of the abelian extensions ofKof degreepby relating the coefficients of the generating polynomials of extensionsL/Kof de-greepto the exponents of generators of the norm groupNL/K(L∗).
This is applied in an algorithm for the construction of class fields of degree pm, which yields an algorithm for the computation of
class fields in general.
SebastianPauli
Department of Mathematics and Statistics University of North Carolina Greensboro Greensboro, NC 27402, USA
E-mail:[email protected]
