UC Santa Cruz

This dissertation introduces and studies the p-adic analytic Brauer group, a cohomological invariant that provides an analytic perspective on the classical theory of central simple algebras over p-adic fields. For a p-adic field F with absolute Galois group GF, we define this invariant as the continuous Galois cohomology group H2(GF, CxF), where CF is the completion of an algebraic closure of F. Our central result establishes that this analytic group is canonically isomorphic to the prime-to-p part of the classical Brauer group of F. Consequently, while division algebras of p-power index over F are trivial in H2(GF, CxF), we provide an analytic construction for every such division algebra by demonstrating that they arise from CF-semilinear representations.