Let E be the splitting field of x4 + 1 over Q. Find Gal(E/Q). Find allsubfields of E. Find the automorphisms of E that have fixed fieldsQ(√2), Q(√-2 ), and Q(i). Is there an automorphism of E whosefixed field is Q?

Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Letv1 =1+x+x2 +x3, v2 =1+2x+3x2, v3 =2+6x, v4 =6. Are the vectors v1, v2, v3, v4 linearly independent over each of K and explain why. (i) K=R; (ii) K = F3, the field of integers mod 3; (iii) K = F5, the field of integers mod 5.

Exactly one of the following pairs consists of two vector spaces over the field R that are not isomorphic to each other. Which one? The correct answer is (b), but why?

Construct an isomorphism between K and L, where K is the splitting field of x3 − x − 1 and L is the splitting field of x3 − x + 1 over F3