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Mat. Zametki, 2009, Volume 85, Issue 5, Pages 661–670 (Mi mz4673)  

Birational Composition of Quadratic Forms over a Local Field

A. A. Bondarenko

Belarussian State University

Abstract: Let $f(X)$ and $g(Y)$ be nondegenerate quadratic forms of dimensions $m$ and $n$, respectively, over $K$, $\operatorname{char} K\ne 2$. The problem of birational composition of $f(X)$ and $g(Y)$ is considered: When is the product $f(X)\cdot g(Y)$ birationally equivalent over $K$ to a quadratic form $h(Z)$ over $K$ of dimension $m+n$? The solution of the birational composition problem for anisotropic quadratic forms over $K$ in the case of $m=n=2$ is given. The main result of the paper is the complete solution of the birational composition problem for forms $f(X)$ and $g(Y)$ over a local field $P$, $\operatorname{char}P\ne 2$.

Keywords: quadratic form, anisotropic quadratic form, binary quadratic form, birational composition, local field, birational composition, Hilbert symbol

DOI: https://doi.org/10.4213/mzm4673

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English version:
Mathematical Notes, 2009, 85:5, 638–646

Bibliographic databases:

UDC: 513.6
Received: 11.03.2008

Citation: A. A. Bondarenko, “Birational Composition of Quadratic Forms over a Local Field”, Mat. Zametki, 85:5 (2009), 661–670; Math. Notes, 85:5 (2009), 638–646

Citation in format AMSBIB
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