The Hilbert symbol for tamely ramified abelian extensions of 2-adic number fields (Q1086292)

Let \(k\) be a 2-adic number field, and let \(K\) be a finite, tamely ramified Galois extension of \(k\) with Galois group \(G\). Let \(q\) be the number of \(p\)-power roots of unity contained in \(K\). The main result of this paper is the determination of the isometry of the Hilbert symbol (or norm residue symbol), \[ \langle\cdot, \cdot\rangle_K: K^{\times}/K^{\times q} \times K^{\times}/K^{\times q} \to \mu_q, \] as a \(G\)-invariant form in the case \(q=2\) and \(G\) abelian, extending an earlier result of \textit{H. Koch} [Math Nachr. 52, 355--369 (1972; Zbl 0241.12008)].