Ph.D. Preliminary Examination in Algebra

August 31, 2001

1. Let $A$ denote the matrix $$\left(\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 1\\ \hfill 5& \hfill -4& \hfill 3\\ \hfill 3& \hfill -3& \hfill 2\end{array}\right),$$ and let $f$ be the $Q$-linear endomorphism of the vector space $Q^3$ given by $f\left(x\right)=Ax$. Find the dimension of the quotient vector space $Q^3/\mathrm{Image}\left(f\right)$.

2. Let $N$ be a normal subgroup of a group $G$ with finite index $\left[G:N\right]=k$. Show that $g^k\in N$ for each element $g\in G$.

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 $$\left(2+\sqrt{-5}\right)\left(2-\sqrt{-5}\right)=\left(3\right)\left(3\right)$$ bear on the question of whether or not the ring $$Z\left[t\right]/\left(t^2+5\right)Z\left[t\right]$$ is a principal ideal domain? (In this $Z$ denotes the ring of integers, and $Z\left[t\right]$ denotes the ring of polynomials in one variable over $Z$. ) Explain your answer.

5. Let $M$ be the $4\times 4$ matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill -1& \hfill 1& \hfill -1\\ \hfill 0& \hfill -1& \hfill 0& \hfill 0\end{array}\right)\text{.}$$ 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.