  The fractional Galois ideal of [Victor P. Snaith, Stark's 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 Stark's
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.
