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

## Practice Final

## Due: Monday, November 27.

**1**. Let F be a contravariant functor from the category of rings
to the category of sets.
Let

f: R--> Z

be any homomorphism of a ring R to the ring Z of integers.
Show that
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

x^{n} + y^{n} = 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
a ring.

**3**. Let G = GL_{3}(F_{2}) be the group of
invertible 3 x 3 matrices with entries in the field with two elements,
acting on the vector space V=F_{2}^{3} of column vectors
in the usual way (by left multiplication).
Let X_{1} be the set of non-zero vectors in V, and
let X_{2} 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 X_{1} and X_{2} 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 X_{1} and
X_{2} over the field of complex numbers are isomorphic
as linear representations of G.

**4**. Write down the class equation for the alternating group
A_{5} on 5 letters.
Show that A_{5} 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
s_{a} of Q(z) determined by the rule

s_{a}(z) = z^{a}.

**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

s_{a}(X(g)) = X(g^{a}).

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
of g.

**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
question 6.)

**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
over Q.