Trace Forms and Stickelberger Relations

Abstract

Let N/ K be a tamely ramified abelian extension of odd degree and let G = Gal( N/ K). This paper studies the equivariant isometry class of the trace form t N/ K restricted to the square root of the inverse different A N/ K . The failure of A N/ K to admit an orthonormal normal basis is measured by an invariant ρ N/ K in the unitary class group UCl( O KG ). This paper shows that for Kummer extensions of odd prime degree, there are Stickelberger-like conditions that determine when a class in UCl( O KG ) can be realized as the ρ-invariant of some tame G-extension.