Abstract. Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant.

Abstract. The Brumer-Stark conjecture deals with abelian extensions of number fields and predicts that a group ring element, called the Brumer-Stickelberger element constructed from special values of L-functions associated to the extension, annihilates the ideal class group of the extension under consideration. Moreover it specifies that the generators obtained have special properties. The aim of this article is to propose a generalization of this conjecture to non-abelian Galois extensions that is, in spirit, very similar to the original conjecture.

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Abstract:We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding ζ-functions, using syntomic regulators and p-adic Lfunctions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of the conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in various other cases.

Abstract. As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the l-part of the Brumer-Stark conjecture for all odd primes l = p with F belonging to a wide class of base fields. In the same setting, we study the 2-part and p-part of Brumer-Stark with no special restriction on F and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases. Overview and resultsAn important conjecture due to Brumer predicts that specific group ring elements constructed from the values of partial zeta-functions at s = 0 annihilate the ideal class groups of certain number fields. Recent progress has been made on this conjecture ([Gr1], [Wi]) and this will be used to obtain new results on the related conjecture of Brumer-Stark where progress thus far has been more restricted (see Section 1 of [RT] for the present status of the Brumer-Stark conjecture). The setting for these conjectures is the following: K/F is a relative Galois extension, with G = Gal(K/F ) abelian, K totally complex, and F totally real. Both conjectures predict that certain elements of Z[G] annihilate the ideal class group Cl K of K.Our results fall naturally into two parts. The first part (Sections 1-3) presents our theoretical results, which we now briefly state:(1) (Probably well known) There is the following localization principle: The Brumer-Stark conjecture (BS) holds for K/F if and only if, for all prime numbers l, an "l-primary analog" (BS) l holds for K/F .(2) (Semi-simple case) If F is an abelian extension of Q with the l-part of Gal(F/Q) cyclic, K is a CM field, and l = 2 is prime to the order of G, then (BS) l holds. For example, given this setup, if G is abelian of order 2p, p an odd prime, then it suffices to prove (BS) 2 and (BS) p in order to establish (BS) for K/F . (The condition that F be abelian over Q can be relaxed somewhat; see Proposition 1.3 for details.) (3) If G is abelian of order 2p, p an odd prime, then (BS) p holds unless K/F is of type or . Here means that K contains a primitive p-th root of unity ζ p and