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3 papers
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 $\overline{a},\overline{b_1},\dots,\overline{b_n}\in k^*/{k^*}^2$ 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
$K(\sqrt{b_1},\dots,\sqrt{b_n})/K$ is not 4-excellent, and the quadratic form
$\langle uv,-u,-v,a\rangle$ provides a corresponding counterexample.
4) The division algebra
$A=D\otimes_E (a,t_0)\otimes_E (b_1,t_1)\dots\otimes_E (b_n,t_n)$ does not decompose into a tensor product of two nontrivial central simple algebras over
$E$, where
$E=K((t_0))((t_1))\dots((t_n))$ is the Laurent series field in variables
$t_0,t_1,\dots,t_n$.
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.
UDC:
512.552,
512.647.2,
512.77 Received: 09.11.2006