A characterization of tame Hilbert-symbol equivalence. (English) Zbl 1024.11022

Let \(K\) and \(L\) be number fields and \(\Omega _K\) and \(\Omega _L\) be the sets of places of \(K\) and \(L\), respectively. The fields \(K\) and \(L\) are said to be tamely Hilbert-symbol equivalent if there are an isomorphism \(t\) from \(K^\ast /{K^\ast }^2\) onto \(L^\ast /{L^\ast }^2\) and a bijection \(T\) from \(\Omega _K\) onto \(\Omega _L\) with the following properties:
(a) \((a,b)_\wp =(ta,tb)_{T\wp }\) for each \(a,b\in K^\ast /{K^\ast }^2\) and each \(\wp \in \Omega _K\), where \((a,b)_\wp \) is Hilbert symbol,
(b) \(\text{ord}_\wp a \equiv \text{ord}_{T\wp } ta\pmod 2\) for each \(a\in K^\ast /{K^\ast }^2\) and every finite place \(\wp \) of \(K\).
The author recalls the definition of the Knebusch-Milnor sequence for a number field by means of the Witt groups of this field, its ring of integers, and the residue class field of its \(\wp \)-adic completion. The presented results give the following characterization of tamely Hilbert-symbol equivalence: “Two number fields are tamely Hilbert-symbol equivalent if and only if they have isomorphic Knebusch-Milnor exact sequences”.


11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E12 Quadratic forms over global rings and fields
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