Eliyahu Matzri

Vanishing of Massey Products and Brauer Groups

Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Minayc, and Tan, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Non-cyclic algebras with n-central elements

We construct, for any prime p, a non-cyclic central simple algebra of degree p2 with p2-central elements. This construction answers a problem of Peter Roquette.

ALL DIHEDRAL DIVISION ALGEBRAS OF DEGREE FIVE ARE CYCLIC

In 1982 Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree 2n of the center, n odd, is in fact cyclic. The proof requires roots of unity of order n in the center. We show that for n = 5, this assumption can be removed. It then follows that 5 Br(F), the 5-torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev (1983) on the 2 and 3 torsion parts.

Diophantine and cohomological dimensions

We give explicit linear bounds on the p-cohomological dimension of a field in terms of its Diophantine dimension. In particular, we show that for a field of Diophantine dimension at most 4, the 3-cohomological dimension is less than or equal to the Diophantine dimension.