Math 6320, Assignment 2 Due in class: Tuesday, February 16

```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 &gt; 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
+ + &middot; &middot; &middot; + ∈ 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].
```