Ph.D. Preliminary Examination in Algebra

January 25, 1999

1. Let $G$ be a finite group and $\mathrm{Perm}\left(G\right)$ the group of permutations of $G$ viewed as a set.

   1. Show that the map $$\lambda :G\to \mathrm{Perm}\left(G\right)$$ that is defined by $\lambda \left(\sigma \right)\left(\tau \right)=\sigma \circ \tau$ is a group homomorphism.

   2. Show that the map ${\rho }_1$ defined by ${\rho }_1\left(\sigma \right)\left(\tau \right)=\tau \circ \sigma$ is a homomorphism if and only if $G$ is an abelian group.

   3. Show that the map $\rho$ defined by $\rho \left(\sigma \right)\left(\tau \right)=\tau \circ {\sigma }^{-1}$ is a homomorphism for every group $G$.

2. Let $A$ be an $n\times n$ matrix in a field $K$, let $c\left(t\right)$ be the characteristic polynomial of $A$, and let $m\left(t\right)$ be the minimal polynomial of $A$. Show that $m\left(t\right)$ divides $c\left(t\right)$ in the polynomial ring $K\left[t\right]$.

3. Show that the alternating group $A_4$ has no subgroup of index $2$.

4. Let $f\left(x\right)=x^5-2$ in $Q\left[x\right]$, and let $K$ be the splitting field of $f\left(x\right)$ over Q.

   1. Find generators for $K$ as a $Q$-algebra.

   2. Find the Galois group $G$ of $K$ over Q.

   3. For each subgroup $H$ of $G$ describe the subfield of $K$ which corresponds to $H$ under the “fundamental correspondence of Galois theory”.

5. Show that if a finite ring $R$ admits an injective (ring) homomorphism from a field, then the number of elements of $R$ must be a power of a prime number.

6. Let $R$ be a commutative ring, $H$ a commutative $R$-algebra, and $I$ an ideal in $H$. Show that $$H/I{\otimes }_{R}H/I\cong \frac{H{\otimes }_{R}H}{I{\otimes }_{R}H+H{\otimes }_{R}I}\text{.}$$

7. Let the field $L$ be a (finite) Galois extension of the field $K$. Define $\mathrm{tr}:L\to K$ by $$\mathrm{tr}\left(\alpha \right)=\sum _{\sigma \in G}\sigma \left(\alpha \right)\text{.}$$ Show that this trace map is surjective on $K$.

8. Let $R$ be a ring and $P$ a left $R$-module. Show that the following two statements are equivalent:

   1. $P$ is a direct summand of a finitely-generated free left $R$-module.

   2. There exist $x_1,\dots ,x_n\in P$, and $f_1,\dots ,f_n\in {\mathrm{Hom}}_R\left(P,R\right)$ such that the relation $$x=\sum f_i\left(x\right)x_i$$ holds for all $x\in P$.