Variations on a theme of Stark

Zeta functions and L-functions contain arithmetic information when evaluated at special values, such as s = 0. In the 1970s, Stark conjectured that the derivatives of L-functions at s = 0 can be evaluated by certain algebraic units. Under certain circumstances, these “Stark units” should also produce abelian extensions of number fields. After introducing the First Order Stark Conjecture, we explore an extended version in which no prime splits completely.