On some elements of the Brauer group of a~conic

A. S. Sivatski

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 227-241

Résumé

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.