Algebra Preliminary Exam

August, 2003

1) Let GL_{n}(F_{p}) denote the group of invertible n×n matrices with entries in the field of p elements F_{p} = Z/pZ. Equivalently, GL_{n}(F_{p}) denotes the group under composition of the 1-1 and onto linear transformations from F_{p}^{n} to F_{p}^{n}.

a) Determine the order of GL_{3}(F_{p}).

b) Show that the subgroup U of GL_{3}(F_{p}) consisting of matrices of the form I + N where N is upper triangular with zero diagonal, is a p-Sylow subgroup of GL_{n}(F_{p}).

c) Show that U is nilpotent but not abelian.

2) If G is a group, let Aut(G) be the group of automorphisms of G. Define C : G → Aut(G) to be the function defined by C(g)(x) = gxg^{−1} for g, x in G.

a) Show that C is a homomorphism.

b) Describe the kernel of C.

c) Show that if G = S_{3}, the symmetric group on three symbols, then C is an isomorphism.

3) Define the semi-direct product of two groups as follows: let (G, ∗) and (A, ·) be groups, and α : A → Aut(G) a homomorphism. Then

G ⋊_{α} A = {(g, a) ∈ G × A}

with multiplication:

(g_{1}, a_{1}) · (g_{2}, a_{2}) = (g_{1} ∗ α(a_{1})(g_{2}), a_{1} · a_{2}).

a) Show that if α is trivial, then G ⋊_{α} A ≅ G × A.

b) Find six pairwise non-isomorphic groups of order 42, all of which are semi-direct products.

4) If R is a commutative ring with 1 and I, J are ideals of R so that

I + J = R, then

R/(I ∩ J) ≅ R/I ⊕ R/J.

Show that when R = Z, the integers, this theorem is equivalent to a well-known theorem in elementary number theory. Explain.

5) Find the Galois group over F_{2}, the field of 2 elements, of the irreducible polynomial f(x) = x^{5} + x^{2} + 1.

6) How many similarity classes of matrices with complex entries have characteristic polynomial (x − 1)(x − 2)^{2}(x − 3)^{3}? Explain.

7) Let F be a finite field.

a) Define the characteristic of F.

b) Prove that the characteristic of F must be a prime p.

c) Prove that the cardinality of F must be a power of p.

d) Show that F is the splitting field of some irreducible polynomial.

8) Suppose R is a ring.

a) What does it mean to say that R is a Euclidean Ring?

c) What does it mean to say that R is a Principle Ideal Domain?

c) Prove that every Euclidean Ring is a Principle Ideal Domain.