Further directions (and potential paper topics)
in group/Galois cohomology and central simple algebras

NB: as with the "Beyond Galois" list, Some of these topics are interrelated (even among different sublists); some are interactions of Galois theory with other branches of mathematics and require tools outside Math 250a and its prerequisites. Most of this material is covered in the sources already mentioned: besides the Tate handout, there's Serre's Local fields, Silverman's The Arithmetic of Elliptic Curves, and Lang's Algebra.

More about the general theory

Further manifestations of Galois cohomology

Central simple algebras and the Brauer group