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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
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 a ring.


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 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.