On some elements of the Brauer group of a conic

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements are linearly independent over ℤ/2ℤ. 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 is not 4-excellent, and the quadratic form 〈uv,-u,-v, a〉 provides a relevant counterexample
(4)	the central division algebra A = D ⊗E (a, t0) ⊗E (b1, t1) ⋯ ⊗E (bn, tn) does not decompose into a tensor product of two nontrivial central simple algebras over E, where E = K ((t0))((t1)) … ((tn)) is the Laurent series field in the variables t0, t1, …, tn
(5)	ind A = 2n+1.

In particular, the algebra A provides an example of an indecomposable algebra of index 2ⁿ⁺¹ over a field, the u-invariant and the 2-cohomological dimension of which equal 2ⁿ⁺³ and n + 3, respectively. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 227–241.