Cofinality of Galois cohomology within purely quadratic graded algebras

Let p be a prime number. For a field F containing a root of unity of order p , let H^{\bullet}(F)=H^{\bullet}(F,\mathbb{F}_{p}) be the mod- p Galois cohomology graded \mathbb{F}_{p} -algebra of F . By the Norm Residue Theorem, H^{\bullet}(F) is a purely quadratic graded-commutative algebra, and is therefore determined by the cup product \cup\colon H^{1}(F)\times H^{1}(F)\to H^{2}(F) . We prove that the class of all Galois cohomology algebras H^{\bullet}(F) is cofinal in the class of all purely quadratic graded-commutative \mathbb{F}_{p} -algebras A_{\bullet} , in the following sense: For every A_{\bullet} there exists F such that the bilinear map A_{1}\times A_{1}\to A_{2} , which determines A_{\bullet} , embeds in the cup product bilinear map \cup\colon H^{1}(F)\times H^{1}(F)\to H^{2}(F) .We further provide examples of \mathbb{F}_{p} -bilinear maps which are not realizable by fields F in this way. These are related to recent results by Snopce–Zalesskii and Blumer-Quadrelli–Weigel on the Galois theory of pro- p right-angled Artin groups, as well as to a conjecture by Marshall on the possible axiomatization of quadratic form theory of fields.