Rings of integers and trace forms for tame extensions of odd degree (Q757475)

Set \(\Gamma =Gal(N/K)\) where N/K is a tame Galois extension of number fields of odd degree. Let R denote the ring of integers of N and \(t_ K: N\times N\to K\Gamma\) be the Hermitian form associated with the trace map for N/K. The main goal of this article is to determine the image, under restriction, of the class of the Hermitian pair \((R,t_ K)\) in the Grothendieck group of locally free \(Z\Gamma\)-modules. A characterization is given in terms of generalized Swan modules which generalize the Lagrangian bases that were determined by \textit{P. E. Conner} and \textit{R. Perlis} [A survey of trace forms of algebraic number fields (1984; Zbl 0551.10017)]. The Artin root numbers play a crucial role in the proof of the main theorem. Also, the proof gives rise to the question of a possible converse result: namely, does the class determine the square of the root numbers of all characters of \(\Gamma\) ?