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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”.

##### MSC:
 1.1e+82 Algebraic theory of quadratic forms; Witt groups and rings 1.1e+13 Quadratic forms over global rings and fields
##### Keywords:
Hilbert symbol; Knebusch-Milnor sequence; Witt ring
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