A generalization of Gauss' theorem on the genera of quadratic forms (Q1066191)

Let k be an algebraic number field of degree \(n_ 0\) over \({\mathbb{Q}}\) which contains a primitive nth root of unity (n\(\geq 2)\) and K/k be a cyclic extension of degree n. Let the torus T be \(R_{k/{\mathbb{Q}}}\) applied to the kernel of the norm map \(R_{K/k}(G_ m)\to G_ m/k\). The author reports a formula for the alternating product \(h_ K (h_ k h_ T)^{-1}\). In particular if \(n=\ell\) a prime, \(h_ K (h_ k h_ T)^{-1}=\ell^{n_ 0(\ell -1)+t-e}\). This reduces to a formula of Gauss when \(k={\mathbb{Q}}\).