Trace Forms and Stickelberger Relations Original Research Article

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 tN/K restricted to the square root of the inverse different AN/K. The failure of AN/K to admit an orthonormal normal basis is measured by an invariant ρN/K in the unitary class group UCl(OKG). This paper shows that for Kummer extensions of odd prime degree, there are Stickelberger-like conditions that determine when a class in UCl(OKG) can be realized as the ρ-invariant of some tame G-extension.