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

• (Co)homogical algebra in general, and how group cohomology fits in that context
• Homology of finite groups, Tate cohomology H^r(G,A) [where r may be also a negative integer, giving homology and cohomology together], and the cohomological proof of the triviality of the Brauer group of finite fields (indicated in the second Serre handout)
• Cup products and/or spectral sequences and their uses

Further manifestations of Galois cohomology

• Noncommutative H¹(Gal(K/k),M) and ``twists''; for example, quadratic forms (if M is an orthogonal group), twists and descents for elliptic curves
• The long exact sequence of noncommutative H<sup>1</sup>(G,M), extended to H<sup>2</sup>(G,M') when the normal subgroup M' is commutative; application to the Brauer group
• Cohomological aspects of the Noether (inverse Galois) problem, see Serre's Topics in Galois Theory
• Descents on elliptic curves (see Tate; one goal is relating the cohomological description of ``2-descent'' with the classical descent of Fermat et al.)

Central simple algebras and the Brauer group

• The Brauer group of non-archimedean local fields such as Q_p
• Quaternion algebras over Q and quadratic reciprocity
• Brauer-Severi varieties: geometrical manifestations of skew fields