189-570A: Higher Algebra I

Assignment 4

Due: Monday, October 30.

1. Describe the conjugacy classes in PSL₂(F_p) when p is congruent to 3 modulo 4.

2. Show that the principal series representations of the group PSL₂(F_p) are all irreducible when p is congruent to 3 modulo 4.

3. Let K=Q(t), where t is a root of the equation
t³ + t² + t + 2 = 0.
Write (t²+t+1)(t²+t) and (t-1)^-1 in the form a+bt + c t².

4. Let t be the (unique) positive real number such that t⁴=5.

(a) Show that Q(it²) is normal over Q.

(b) Show that Q(t+it) is normal over Q(it²).

(c) Show that Q(t+it) is not normal over Q.

5. Let K be a field. Prove that every algebraic extension of K is seperable, if and only if K has characteristic 0, or K has characteristic p and every element of K has a pth root in K.

6. Let k be a field of characteristic p and let u, v be indeterminates. Show that:

(a) k(u,v) has degree p² over k(u^p, v^p).

(b) There are infnitely many distinct extensions containing k(u^p,v^p) and contained in k(u,v).

7. Let k be a field of characteristic p, and let f(x) be a polynomial in k[x] of the form x^pⁿ + a_n-1 x^{p^n-1} + a_n-2 x^{p^n-2} + ... + a₁ x^p + a₀ x. Show that f(x) is seperable if and only if a₀ is non-zero. In that case, show that the Galois group of f(x) is isomorphic to a subgroup of the group GL_n(F_p) of n by n invertible matrices with entries in the field with p elements.

8. Let k be a field of characteristic p, and let K be a cyclic Galois extension of k of degree p. Show that K is the splitting field of a polynomial of the form x^p - x -a = 0, for some a in k.

9. Let k be the finite field with p elements (p a prime). Let K=k(x) be the field of rational functions in the variable x. Let G be the group of field automorphisms of K obtained by the mappings
x |--> (ax + b) / (cx+d), with a,b,c,d in k, ad-bc non-zero.

Show that

(a) G is isomorphic to the quotient of GL₂(F_p) by its center, and hence has order p³-p.

(b) The subfield of K fixed by the subgroup U of unipotent elements is K^U = k(x^p-x).

(c) The subfield of K fixed by the Borel subgroup B of upper trinagular matrices is K^B = k(u) where u := (x^p-x)^p-1.

10. With notations as in problem 9, show that the subfield of K fixed by G is equal to k(t), where
t = (x^p²-x)^p+1/ (x^p-x)^p²+1.

Conclude from this that the polynomial
f(x) = (x^p²-x)^p+1 - t (x^p-x)^p²+1

with coefficients in F_p(t) has Galois group equal to G.

It follows from this last problem that the group G arises as the Galois group of a finite extension of the field of rational functions over the field with p elements.