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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

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Abstract. Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

Abstract. A method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian Stark Conjecture for some totally real cubic fields of discriminant less than 50000. The results of these computations are used to provide explicit Hilbert class fields and some ray class fields for the cubic extensions.