189-570A: Higher Algebra I
Due: Monday, November 27.
1. Let F be a contravariant functor from the category of rings
to the category of sets.
f: R--> Z
be any homomorphism of a ring R to the ring Z of integers.
F(f) is injective.
2. Let n be an integer greater than 3. Fermat's Last Theorem for the
exponent n - the assertion that the equation
xn + yn = 1
has no rational solution with x, y non-zero -
is equivalent to the assertion that a certain ring admits no
homomorphism to the ring of rational numbers. Write down such
3. Let G = GL3(F2) be the group of
invertible 3 x 3 matrices with entries in the field with two elements,
acting on the vector space V=F23 of column vectors
in the usual way (by left multiplication).
Let X1 be the set of non-zero vectors in V, and
let X2 be the set of two-dimensional subspaces of V, equipped
with the natural G-action induced from the action of G on V.
Show that X1 and X2 are both transitive G-sets of
cardinality 7, which are NOT isomorphic as G-sets.
Show on the other hand
that the permutation representations attached to X1 and
X2 over the field of complex numbers are isomorphic
as linear representations of G.
4. Write down the class equation for the alternating group
A5 on 5 letters.
Show that A5 is a non-abelian simple group.
5. Let p be a prime number and let z be a primitive pth root of unity.
Show that the field Q(z) generated over Q by z is a Galois extension of
Q of degree p-1.
(Hint: show that z-1 satisfies an irreducible polynomial
of that degree by using the Eisenstein criterion.)
Conclude that for all integers a prime to p, there is an automorphism
sa of Q(z) determined by the rule
sa(z) = za.
6. Let G be a group and let g be an element of G of prime order p.
If X is the character of a representation of g, show that X(g) belongs to the
field Q(z) of problem 5, and that
sa(X(g)) = X(ga).
Show that X(g) is rational for all characters X, if and only
if all the powers of g except for the neutral element
belongs to the conjugacy class of g.
More generally, describe
explicitly the field generated over Q
by the entries of the character table of G
in the column corresponding to the conjugacy class of g.
Give a formula for the degree of this extension,
involving only group theoretic properties
7. Fix a prime p and let G be a simple group containing
a unique conjugacy class of elements of
order p. Show that any non-trivial irreducible
representation of G has dimension at least p-1. (Hint: use the ideas of
8. Let S be a subring of a ring R.
Define the integral closure of S in R.
Show that Z is equal to its integral closure in Q.
9. Prove or disprove: if f(x) is
a monic polynomial with coefficients in Z
and the reduction of f modulo p (viewed as a polynomial with
coefficients in the field with p elements)
is reducible for all p, then f is reducible