Math 6320, Assignment 2 Due in class: Tuesday, February 16 1. Show that the only homomorphism R −→ R is the identity. 2. Let K be a subfield of a field L. If ϕ : L −→ L is a homomorphism that fixes all elements of K, is ϕ necessarily an automorphism of L? 3. For each of the following polynomials, determine the number of distinct roots in F49 . x48 − 1, x49 − 1, x54 − 1. 4. Let K be a field of characteristic p > 0. Set M = K(x, y) and L = K(x p , y p ). (a) Prove that [M : L] = p2 . (b) Prove that M 6= L(γ) for any γ ∈ M. 5. For which integers n does the polynomial 1 + xn x x2 + + · · · + ∈ Q[x] have multiple roots? 1! 2! n! 6. Let K be a field. Let f (x) ∈ K[x] be a monic polynomial of degree n with f (0) 6= 0. Suppose f (x) has n distinct roots in its splitting field, and that the set of roots is closed under multiplication. Determine f (x). √ √ 7. Prove that the extension Q ⊂ Q( 2, 3) is Galois, and compute its Galois group. 8. Determine the Galois group of x4 − 2 ∈ Q[x]. Is it cyclic? Is it abelian? 9. Determine the Galois group of x4 + 4x2 + 2 ∈ Q[x]. 10. Determine the Galois group of x6 + 3 ∈ Q[x].