Abstract

We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark's Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples. In a series of important papers [Stark 71, Stark 75, Stark 76, Stark 80] H. M. Stark developed a body of conjectures relating the values of Artin L-functions at s = 1 (and hence, by the functional equation, their leading terms at s = 0) with certain algebraic quantities attached to extensions of number fields. For example, in the case of Abelian L-functions with a first-order zero at s = 0; the conjectural relation is between the first derivative of the L-functions and the logarithmic embedding of certain units in ray class fields known as Stark units, which are predicted to exist. The use of these conjectures to provide explicit generat...

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