Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

On indecomposable algebras of exponent 2

Abstract. For any n ≥ 3 we give numerous examples of central division algebras of exponent 2 and index 2 n over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of n + 1 quaternion algebras in the Brauer group of the field. Also for any n ≥ 3 and any field k0 we construct an extension F/k0 and a multiquadratic extension L/F of degree 2 n such that for any proper subextensions L1/F and L2/F W(L/F) � = W(L1/F) + W(L2/F), 2Br(L/F) �=2 Br(L1/F) +2 Br(L2/F)

Department of Mathematics-2

Abstract. Let K be a field of characteristic not 2, X a nonsingular projective conic over K, n is a positive integer and ai ∈ K ∗ for 1 ≤ i ≤ n. We investigate elements of 2Br(X), which are sums of quaternion algebras (ai, fi) for some fi ∈ K(X) ∗. An application to a construction of indecomposable division algebras of exponent 2 is given. The main purpose of this paper is to construct indecomposable central division algebras of an arbitrary 2-primary index more than 4 over fields with finite uinvariant (recall that the u-invariant of the field K is the maximal dimension of anisotropic quadratic forms over K). To do this we begin with a result concerning the Brauer group of a conic. Let K be a field of characteristic different from 2, and let X be a nonsingular projective conic over K. Recall that 2Br(X) = ker ( 2BrK(X) ∂ → � K(x) ∗ /K(x) ∗2), where K(X) is the function field of X, x runs over all closed points of X, and K(x) is the residue field at x. Moreover, x∈X ∂x(f, g) = (−1) vx(f)vx(g) f vx(g) g vx(f) ∈ k(x) ∗ /k(x) ∗2, where vx is the discrete valuation related to the point x, and the symbol (f, g) denotes a quaternion algebra. Notice also that res K(X)/K 2BrK ⊂ 2Br(X). From now on we fix a field K of characteristic not 2, and elements a1,...an ∈ K ∗ such that a1,...an ∈ K ∗ /K∗2 are linearly independent over Z/2Z. For any set I ⊂ {1,..., n}, put aI = � ai (if I = ∅, then aI = 1). For any polynomial i∈I f ∈ K[t] denote by l(f) the leading coefficient of f. Key words and phrases. Quadratic form, Brauer group, Quaternion algebra, Conic. The work under this publication was partially supported by Royal society Joint Project”Quadratic forms and central simple algebras under field extensions

Division Algebras Over Rational Function Fields in One Variable

Abstract. Let A be a central simple algebra over the field of rational functions in one variable over an arbitrary field of characteristic different from 2. If the Schur index of A is not divisible by the characteristic and its ramification locus has degree at most 3, then A is Brauer-equivalent to the tensor product of a quaternion algebra and a constant central division algebra D. The index of A is computed in terms of D and the ramification of A. The result is used to construct various examples of division algebras over rational function fields