On the ''wild'' norm residue symbol in an abelian extension
Let k be a finite extension of the field \({\mathbb{Q}}_ p\) of p-adic numbers. Let L be an abelian extension of k of degree \(n>1\). Let \(\mu\) (k) (resp. \(\mu\) (L)) denote the finite group of roots of unity in k (resp. L). Let \(\mu (k)_{wild}\) (resp. \(\mu (L)_{wild})\) denote the p-primary component of \(\mu\) (k) (resp. \(\mu\) (L)). We assume that \(\mu (k)_{wild}\) is nontrivial. \(L^{\odot}\) will denote the group of elements of \(L^{\times}\) of norm 1 over k. For \(x,y\in L^{\times}\), \((x,y)_{wild}\) will denote the ''wild'' norm residue symbol in L. Recall that if \(p^ s=\#\mu (L)_{wild}\), then \((x,y)_{wild}=\theta (y)(x^{1/p^ s})/x^{1/p^ s}\), where \(\theta =\theta_ x: L^{\times}\to Gal(L(x^{1/p^ s})/L)\) is the surjective homomorphism given by local class field theory. The object of this note is to prove the following Theorem: If \(p=2\), assume that the Galois group of L/k is not of exponent 2. Then there exist \(x,y\in L^{\odot}\) such that \((x,y)_{wild}\) is a generator of \(\mu (L)_{wild}.\) In a forthcoming joint paper of the author with M. S. Raghunathan on topological central extensions of SL\({}_ 1\)(D), this theorem has been used to prove the following: Let D be a finite dimensional central division algebra over k. If \(p=2\) and the degree d of D is even, assume that k contains a primitive fourth root of unity and \(d\neq 2\). Then the cohomology group \(H^ 2(SL_ 1(D), {\mathbb{R}}/{\mathbb{Z}})\), based on continuous cochains, contains a subgroup isomorphic to the Pontryagin dual of \(\mu (k)_{wild}\).
scientific article; zbMATH DE number 3920611