Ph.D. Preliminary Examination in Algebra

August 31, 2001

1. Let A denote the matrix

( \begin{matrix}1 & −2 & 1 \\ 5 & −4 & 3 \\ 3 & −3 & 2 \end{matrix} ) ,

and let f be the Q-linear endomorphism of the vector space Q^{3} given by f(x) = Ax. Find the dimension of the quotient vector space Q^{3}/Image(f).

2. Let N be a normal subgroup of a group G with finite index [G : N] = k. Show that g^{k} ∈ N for each element g ∈ G.

3. Why must the number of elements in a finite field always be the power of some prime?

4. Does the existence of the relationship

(2 + \sqrt{−5}) (2 − \sqrt{−5}) = (3) (3)

bear on the question of whether or not the ring

Z[t]/(t^{2} + 5)Z[t]

is a principal ideal domain? (In this Z denotes the ring of integers, and Z[t] denotes the ring of polynomials in one variable over Z. ) Explain your answer.

5. Let M be the 4 × 4 matrix

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & −1 & 1 & −1 \\ 0 & −1 & 0 & 0 \end{matrix} ) .

Find the characteristic and minimal polynomials of M when it is regarded as a matrix over the field C of complex numbers.

6. What is the Galois group of the polynomial x^{4} + 1 over the field Q of rational numbers?

7. Prove over any commutative ring (with 1) that two isomorphic free modules of finite rank must have the same rank.

8. Find the group of all automorphisms of the symmetric group on 3 letters.