DEPARTMENT OF MATHEMATICS & STATISTICS

Preliminary Ph.D. Examination in Algebra

September 2, 1993

1.  Determine the number of 3 × 3 invertible matrices in a finite field having q elements.

2.

Can a line segment with length equal to the positive real fifth root of 2 be constructed (given unit length) in a finite number of steps using straightedge and compass? Explain.

3.  Let A be a commutative ring (with multiplicative identity).

(a)

Let M be an A-module. Let R = End(M) be the set of endomorphisms of M (i.e., the set of A-module homomorphisms M → M ). Define operations on R that make R an A-algebra (i.e., a ring with compatible A-module structure).

(b)

Is the set of ring endomorphisms (as opposed to A-module endomorphisms) of the ring A a ring?

(c)

When A = Z , the ring of integers, find the endomorphism ring of the Z-module Z ⊕ Z/4Z.

4.  Show that any group of order 20 has a non-trivial proper normal subgroup.

5.

Prove that a finitely-generated torsion-free module over a principal ideal domain is necessarily free.

6.

Determine the isomorphism class of each of the Sylow subgroups of the alternating group A_{5}, the group of “even” permutations of a set of cardinality 5.

7.

Let ζ be a primitive 7th root of unity in the field of complex numbers, let K = Q(ζ), and H = Q(α), where α = cos(2π/7) and Q denotes the field of rational numbers.

(a) Show that H = K ∩ R, where R is the field of real numbers.
(b) Prove that K and H are both normal extensions of Q.
(c) Determine the Galois groups Gal(K : Q), Gal(H : Q), and Gal(K : H).

8.

For p a prime the ring of p-adic integers Z_{p} is defined to be the inverse limit of the unique ring homomorphisms

… → Z/p^{n}Z → … → Z/p^{2}Z → Z/pZ .

Let π denote the canonical ring homomorphism Z_{p} → Z/pZ.

(a)

Show that an element of Z_{p} is invertible in Z_{p} if and only if its image under π is non-zero.

(b) Show that the kernel of π is a maximal ideal of Z_{p}.

(c)

Show that any proper ideal in Z_{p} is contained in the kernel of π. (Hence, ker(π) is the only maximal ideal.)

(d) Show that the kernel of π is a principal ideal.

(e)

Give another (non-isomorphic) example of a ring having a unique maximal ideal that is principal.