The fractional Galois ideal of [Victor P. Snaith, Starks conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419--448] is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected
to provide non-trivial annihilators for higher K-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Starks conjectures, and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.
We propose a candidate, which we call the fractional Galois ideal after Snaiths fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering in
formation about those $L$-functions of the extension which are non-zero at the special point $s = 0$, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include $L$-functions with a simple zero at $s = 0$, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this article deals with $L$-functions of arbitrary order of vanishing at $s = 0$, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Buyukboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.
The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups. In [Victor P. Snaith, Starks conjecture and new Stickelber
ger phenomena, Canad. J. Math. 58 (2) (2006) 419--448], Snaith introduces canonical Galois modules hoped to appear in annihilator relations generalising and improving those involving Stickelberger elements. In this paper we study the first of these modules, corresponding to the classical Stickelberger element, and prove a connection with the Stark units in a special case.
