Math 6320, Assignment 5
Due in class: Thursday, April 21
In the problems below, K denotes a field.
1. Prove that F4 (x) is Galois over F4 (x4 + x), and compute the Galois group. Determine the intermediate fields,
and show that each is rational over F4 .
2. Consider the automorphisms τ and σ of L = K(x) that fix K and satisfy τ(x) = 1/x and σ (x) = 1 − 1/x.
(a) Determine the group G = hτ, σ i.
(b) Prove that the fixed field LG is generated over K by
(x2 − x + 1)3
.
x2 (x − 1)2
3. Let w, x, y, z be variables over a field K.
(a) Compute the transcendence degree of K(x2 yz, xy2 z, xyz2 ) over K.
(b) Compute the transcendence degree of K(wy, wz, xy, xz) over K.
4. Let R be a Noetherian ring, and let ϕ : R −→ R be a surjective ring homomorphism. Is ϕ an isomorphism?
5.
(a) Suppose f (x) ∈ K[x] has positive degree. Prove that K[x] is a finitely generated K[ f ]-module.
(b) Let R be a subring of K[x] that contains K. Prove that R is Noetherian.
6. Let V = {(t,t 2 , . . . ,t n ) | t ∈ C}. Show that V is an algebraic set in Cn .
7. Let K be an algebraically closed field, and L an extension field. If polynomials f1 , . . . , fm in K[x1 , . . . , xn ] have a
common root in Ln , prove they have a common root in K n .
8. Let m be a maximal ideal of R[x, y] containing x2 + y2 + 1. What is the quotient R[x, y]/m?
9. Let R = K[x1 , . . . , xn ] be a polynomial ring over a finite field K.
(a) Given p ∈ K n , construct f ∈ R with f (p) = 1 and f (q) = 0 for all q ∈ K n r {p}.
(b) Given g : K n −→ K, construct f ∈ R with f (p) = g(p) for all p ∈ K n .
(c) Prove that each subset V ⊆ K n is the zero set of some f ∈ R, i.e., an algebraic set.
10. Suppose K is not algebraically closed. Prove that each algebraic set in K n is the zero set of a single polynomial.