# 189-570A: Higher Algebra I

## Due: Monday, October 30.

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

2. Show that the principal series representations of the group PSL2(Fp) 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(up, vp).

(b) There are infnitely many distinct extensions containing k(up,vp) 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 xpn + an-1 xpn-1 + an-2 xpn-2 + ... + a1 xp + 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 GLn(Fp) 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 xp - 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(Fp) by its center, and hence has order p3-p.

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

(c) The subfield of K fixed by the Borel subgroup B of upper trinagular matrices is KB = k(u) where u := (xp-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 = (xp2-x)p+1/ (xp-x)p2+1.

Conclude from this that the polynomial
f(x) = (xp2-x)p+1 - t (xp-x)p2+1

with coefficients in Fp(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.