## 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
Full Text:

### References:

 [1] P. E. Conner R. Perlis, and K. Szymiczek: Wild sets and 2-ranks of class groups. Acta Arithmetica 79 (1997), 83-91. · Zbl 0880.11039 [2] A. Czogala: On integral Witt equivalence of algebraic number fields. Acta Math. Inform. Univ. Ostrav. 4 (1996), 7-21. · Zbl 0870.11022 [3] T. Y. Lam: The Algebraic Theory of Quadratic Forms. Mathematics Lecture Note Series, Benjamin/Cummings Publ. Co., Reading, Mass. 1973. · Zbl 0259.10019 [4] J. Milnor, D. Husemoller: Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag, Berlin Heidelberg New York 1973. · Zbl 0292.10016 [5] R. Perlis K. Szymiczek P. E. Conner, and R. Litherland: Matching Witts with global fields. Contemporary Mathematics 155 (1994), 365-387. · Zbl 0807.11024 [6] W. Scharlau: Quadratic and Hermitian Forms. Grundlehren der mathematischen Wissenschaften 270, Springer-Verlag, Berlin Heidelberg New York Tokyo 1985. · Zbl 0584.10010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.