The tame Hilbert symbol via K-theory and central extensions

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Let K be a mixed characteristic local field whose residue field has cardinality q, and let n be an integer dividing q−1. In the first part of this document, we construct a K-theoretic enhancement of the n-th power residue symbol K××K×→μn. In the second part we construct central extensions of GLm(K) by μn and we express the n-th power residue symbol in terms of a symbol defined using the extension obtained for m=1. Our constructions both rely on the study of finite free pointed μn-sets.