Let p be a prime number, F a field containing a root of unity of order p , and G_F the absolute Galois group. Extending results of Hopkins, Wickelgren, Mináč and Tân, we prove that the triple Massey product H^1(G_F)^3\to H^2(G_F) contains 0 whenever it is non-empty. This gives a new restriction on the possible profinite group structure of G_F .

Abstract. We bound the symbol length of elements in the Brauer group of a field K containing a C m field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a C m field F . In particular, for a C m field F , we show that every F central simple algebra of exponent p t is similar to the tensor product of at most len(p t , F ) ≤ t(p m−1 − 1) symbol algebras of degree p t . We then use this bound on the symbol length to show that the index of such algebras is bounded by (p t ) (p m−1 −1) , which in turn gives a bound for any algebra of exponent n via the primary decomposition. Finally for a field K containing a C m field F , we show that every F central simple algebra of exponent p t and degree p s is similar to the tensor product of at most len(p t , p s , K) ≤ len(p t , L) symbol algebras of degree p t , where L is a C m+edL(A)+p s−t −1 field.

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Let p be a prime number, F a field containing a root of unity of order p, and G F the absolute Galois group. Extending results of Hopkins, Wickelgren, Mináč and Tân, we prove that the triple Massey product H 1 (G F ) 3 → H 2 (G F ) contains 0 whenever it is nonempty. This gives a new restriction on the possible profinite group structure of G F .A main problem in modern Galois theory is to understand the grouptheoretic structure of absolute Galois groups G F = Gal(F sep /F ) of fields F , that is, the possible symmetry patterns of roots of polynomials. General restrictions on the possible structure of the profinite group G F are rare: By classical results of Artin and Schreier, the torsion in G F can consist only of involutions. In addition, the celebrated work of Voevodsky and Rost ([Voe03], [Voe11]) identifies the cohomology ring H * (G F ) = H * (G F , Z/m) with the mod-m Milnor K-ring K M * (F )/m, assuming existence of m-th roots of unity. In particular, the graded ring H * (G F ) is generated by its degree 1 elements, and its relations originate from the degree 2 component. This can be used to rule out many more profinite groups from being absolute Galois groups of fields ([CEM12], [EM11b]). In fact, the Artin-Schreier restriction about the torsion also follows from the latter results [EM11b, Ex. 6.4(2)].Very recently, a remarkable series of works by Hopkins, Wickelgren, Mináč and Tân indicated the possible existence of a new kind of general restrictions on the structure of absolute Galois groups, related to the differential graded algebra C * (G F ) = C * (G F , Z/m) of continuous cochains on G F . The interplay between C * (G F ) and its cohomology algebra H * (G F ) gives rise to external operations on H * (G F ), in addition to its ("internal") ring structure with respect to the cup product, notably, the n-fold Massey

Abstract. 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, Mináč, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.