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 Zap. Nauchn. Sem. POMI, 2006, Volume 338, Pages 227–241 (Mi znsl175)

On some elements of the Brauer group of a conic

A. S. Sivatski

Saint-Petersburg State Electrotechnical University

Abstract: The main purpose of this paper is to strenghen the author's results in articles [7] and [8]. Let $k$ be a field of characteristic $\ne 2$, $n\ge 2$. Suppose that elements $\overline{a},\overline{b_1},…,\overline{b_n}\in k^*/{k^*}^2$ are linearly independent over $\mathbb Z/2\mathbb Z$. We construct a field extension $K/k$ and a quaternion algebra $D=(u,v)$ over $K$ such that
1) The field $K$ has no proper extension of odd degree.
2) The $u$-invariant of $K$ equals 4.
3) The multiquadratic extension $K(\sqrt{b_1},…,\sqrt{b_n})/K$ is not 4-excellent, and the quadratic form $\langle uv,-u,-v,a\rangle$ provides a corresponding counterexample.
4) The division algebra $A=D\otimes_E (a,t_0)\otimes_E (b_1,t_1)…\otimes_E (b_n,t_n)$ does not decompose into a tensor product of two nontrivial central simple algebras over $E$, where $E=K((t_0))((t_1))…((t_n))$ is the Laurent series field in variables $t_0,t_1,…,t_n$.
5) $\operatorname{ind}A=2^{n+1}$.
In particular, the algebra $A$ provides an example of an indecomposable algebra of index $2^{n+1}$ over a field, whose $u$-invariant and 2-cohomological dimension equal $2^{n+3}$ and $n+3$, respectively.

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English version:
Journal of Mathematical Sciences (New York), 2007, 145:1, 4823–4830

Bibliographic databases:

UDC: 512.552, 512.647.2, 512.77

Citation: A. S. Sivatski, “On some elements of the Brauer group of a conic”, Problems in the theory of representations of algebras and groups. Part 14, Zap. Nauchn. Sem. POMI, 338, POMI, St. Petersburg, 2006, 227–241; J. Math. Sci. (N. Y.), 145:1 (2007), 4823–4830

Citation in format AMSBIB
\Bibitem{Siv06} \by A.~S.~Sivatski \paper On some elements of the Brauer group of a~conic \inbook Problems in the theory of representations of algebras and groups. Part~14 \serial Zap. Nauchn. Sem. POMI \yr 2006 \vol 338 \pages 227--241 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl175} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2355336} \zmath{https://zbmath.org/?q=an:1120.16019|1113.11025} \elib{https://elibrary.ru/item.asp?id=9305297} \transl \jour J. Math. Sci. (N. Y.) \yr 2007 \vol 145 \issue 1 \pages 4823--4830 \crossref{https://doi.org/10.1007/s10958-007-0315-y} \elib{https://elibrary.ru/item.asp?id=13539962} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34547573024} 

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This publication is cited in the following articles:
1. Sivatski A.S., “On the Brauer group complex for a multiquadratic field extension”, J. Algebra, 323:2 (2010), 336–348
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