The First Order Abelian Stark Conjecture establishes a connection between analytic and algebraic number theory. In the 1970's, Harold Stark [10] conjectured the existence of certain algebraic units which evaluate the first derivatives of abelian L-functions at s = 0. Furthermore, certain roots of these algebraic units explicitly generate maximal abelian extensions of the base field. Hence, Stark conjecturally provides an answer to Hilbert's Twelfth Problem, which asks for a method of constructing abelian extensions of number fields using analytic functions.; The First Order Abelian Stark Conjecture requires that all the L-functions of a given extension K/k vanish at s = 0. This requirement has traditionally been satisfied by supposing that some prime of k splits completely in the extension K/k. However, there are other situations where all L-functions vanish at s = 0. The main goal of this thesis is to extend the conjecture to this more general setting.; After setting up notation and motivation for the Extended First Order Abelian Stark Question, we will state the question and reduce it to proving that the Stark units from intermediate fields are certain powers of elements in the top field. We prove that the reduction is satisfied under certain conditions. Finally we provide some explicit examples which test the boundaries of the extension.
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