**189-570A:** Higher Algebra I

## Assignment 4

## Due: Monday, October 30.

**1**.
Describe the conjugacy classes in
**PSL**_{2}(**F**_{p})
when
p is congruent to 3 modulo 4.

**2**. Show that the principal series representations of the group
**PSL**_{2}(**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

t^{3} + t^{2} + t + 2 = 0.

Write (t^{2}+t+1)(t^{2}+t) and
(t-1)^{-1} in the form a+bt + c t^{2}.

**4.**
Let t be the (unique) positive real number such that t^{4}=5.

(a) Show that Q(it^{2}) is normal over Q.

(b) Show that Q(t+it) is normal over Q(it^{2}).

(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 p^{2}
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^{pn-1} +
a_{n-2} x^{pn-2} + ... +
a_{1} x^{p} + a_{0} x.
Show that f(x) is seperable if and only if a_{0} 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
GL_{2}(F_{p}) by its center, and hence has
order p^{3}-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.