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1. p(x)=x^{7}+11 over Q(a), R.

where a is 7-th root of unity. What are Galouis groups?

For the 1st case I got Z_{7}, second not sure. need hint for that

2. need hint. I know it is easy: M is an R-module. Show that Hom_{R}(R,M)[tex]\cong[/tex]M.

3. Spse that I is an ideal of R such that I^{k}=0 for some k>0 integer. Let M, N be R-modules and let [tex]\phi[/tex]:M->N be an R-module hom. Prove that if induced map [tex]\bar{\phi}[/tex]:M/IM->N/IN is surjective, then [tex]\phi[/tex] is surjective.

4. show that 2[tex]\otimes[/tex]1 [tex]\neq[/tex]0 in 2Z[tex]\otimes[/tex]Z/2Z.