189-570A: Higher Algebra I
Due: Monday, October 30.
Describe the conjugacy classes in
p is congruent to 3 modulo 4.
2. Show that the principal series representations of the group
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.
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.
(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
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
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.
G is isomorphic to the quotient of
GL2(Fp) by its center, and hence has
(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
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/
Conclude from this that the polynomial
f(x) = (xp2-x)p+1 - t
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.