The Schur subgroup of the Brauer group of a local field (Q1107503)

Let \(K\) be a finite extension of \({\mathbb Q}_ p\); i.e., \(K\) is a local field; and let \(K_ c=K\cap {\mathbb Q}^ c_ p\), where \({\mathbb Q}^ c_ p\) means the cyclotomic closure of \({\mathbb Q}_ p\), i.e. the field generated over \({\mathbb Q}_ p\) by adjunction of all roots of unity. The author proves that the Schur group \(S(K)\cong \text{tor}(G({\mathbb Q}^ c_ p/K_ c))\), the torsion subgroup of the Galois group of \({\mathbb Q}^ c_ p\) over \(K_ c\). Although this result is known in the literature (especially in the dyadic case, by several authors), the author makes a case for his quite different approach and methods of proof. In the non-dyadic case for example his proof uses the cup product pairing and the norm residue symbol, a different set of techniques than in previous proofs. It should be noted that \textit{F. DeMeyer} and the reviewer have generalized the concept of the Schur group to the commutative ring setting in: (i) their paper in ``Orders and their applications'', Proc. Conf., Oberwolfach 1984, Lect. Notes Math. 1142, 205--210 (1985; Zbl 0582.13003); (ii) J. Pure Appl. Algebra 35, 117--122 (1985; Zbl 0555.13003); and (iii) in ``Invariants of group rings'', J. Algebra 118, No. 2, 336--345 (1988; Zbl 0665.16009)).