Hilbert’s theorem 90 and generalizations

These notes are based on the Number Theory seminar I gave at Purdue University on September 19, 2013 titled “Hilbert’s theorem 90 and generalization”. The proof of Hilbert’s 90 is taken from an answer I found on MathOverflow at http://mathoverflow.net/a/21117. In the seminar, Professor Goins asked an interesting question about the isomorphism classes of rank one tori which I have blogged here. 1 Non-abelian cohomology General abstract nonsense says, in an abelian category a half-exact functor F gives rise to a long exact sequence of (co)homology. In other words, whenever 0→ A→ B → C → 0 is an exact sequence, giving rise to the exact sequence 0→ F(A)→ F(B)→ F(C); then we have 0→ F(A)→ F(B)→ F(C)→ H(A)→ · · · .