Math 514b Algebraic Number Theory Spring 1996

Short exam on group cohomology:

1. Let G = Z/pZ. Recall the definition of the Herbrand quotient h(A) of a G-module A. Compute h(A) for A = Z, A = Q/Z, and A = Z/pZ, all with G acting trivially.

2. Let V = R^{2} with the standard bilinear form ⟨, ⟩. Show that the space V′ = R^{2} with the bilinear

form ⟨, ⟩′ whose matrix is ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) is a twisted form of V. Compute A = Aut_{C}(V, ⟨, ⟩) and a cocycle

representing the class in H^{1}(Gal(C/R), A) corresponding to (V′, ⟨, ⟩′).

3. Let S_{3} act on Z via the sign function: gn = sgn(g)n. Compute H^{1}(S_{3}, Z).

4. Explain how S_{3} can be viewed as an extension of Z/2Z by Z/3Z. Compute the class in H^{2}(Z/2Z, Z/3Z) corresponding to the extension you wrote down. (Optional: Is this class uniquely determined by S_{3}?)