Ph.D. Preliminary Examination in Algebra

June 4, 1999

1. Let $A$ be an $n\times n$ matrix with entries in the field $C$ of complex numbers that satisfies the relation $A^2=A$. Show that $A$ is similar to a diagonal matrix which has only $0$'s and $1$'s along the diagonal.

2. Furnish examples of the following:

   1. A finite group that is solvable but not abelian.

   2. A finite group whose center is a proper subgroup of order $2$.

   3. A nested sequence of finite groups $G,H,K$ with $H$ a normal subgroup of $G$ and $K$ a normal subgroup of $H$ such that $K$ is not a normal subgroup of $G$.

3. Let $p$ be the polynomial $p\left(t\right)=t^5+t^2+1$ regarded as an element of the ring $A=F_2\left[t\right]$ of polynomials with coefficients in the field $F_2$ of two elements. Show that $p$ is irreducible, and then find a polynomial of degree at most $4$ with the property that its residue class modulo the ideal $pA$ generates the entire multiplicative group of units in the quotient ring $A/pA$.

4. Let $G$ be a finite group of order $N$, and let $n$ be a positive integer that divides $N$. Do one of the following:

   1. Prove that if $G$ is abelian, then $G$ contains a subgroup of order $n$.

   2. Find an example of $G,N,n$ as above where $G$ has no subgroup of order $n$.

5. Show that every group of order $30$ contains a normal cyclic subgroup of order $15$.

6. Let $F$ be the field $Q\left(i\right)$ where $i=\sqrt{-1}\in C$, and let $E$ be the splitting field over $F$ of the polynomial $f\left(t\right)=t^4-5$. Find:

   1. the extension degree $\left[E:F\right]$.

   2. the group ${\mathrm{Aut}}_F\left(E\right)$ of all automorphisms of $E$ that fix $F$.

7. Let $F_2$ be the field of $2$ elements, and let $R$ be the commutative ring $$R=F_2\left[t\right]/t^3{F}_2\left[t\right].$$

   1. How many elements does $R$ contain?

   2. What is the characteristic of $R$ ?

   3. Find all ring homomorphisms $R\to R$.

8. Let $a,b,c,d$ be elements of a field $F$, let $A,B,C,D$ be $n\times n$ matrices over $F$, and let $$m=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right) and M=\left(\begin{array}{cc}\hfill A& \hfill B\\ \hfill C& \hfill D\end{array}\right)\text{.}$$ If $\lambda :F^2\to F^2$ and $\Lambda :F^{2n}\to F^{2n}$ denote the linear endomorphisms corresponding (relative to standard coordinates) to $m$ and $M$, respectively, then to what linear endomorphism that may be constructed from $\lambda$ and $\Lambda$ may one relate the $4n\times 4n$ (Kronecker product) matrix $$\left(\begin{array}{cccc}\hfill aA& \hfill bA& \hfill aB& \hfill bB\\ \hfill cA& \hfill dA& \hfill cB& \hfill dB\\ \hfill aC& \hfill bC& \hfill aD& \hfill bD\\ \hfill cC& \hfill dC& \hfill cD& \hfill dD\end{array}\right)\text{?}$$ Explain your answer.