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\centerline{\bf{Ph.D. Preliminary Examination}}

\centerline{\bf{ALGEBRA}}

\centerline{\bf{Fall 1995}}

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\item{1.} Let $K$ be a field, $L$ a field extension of $K$.  An element $\alpha$ in $L$ is algebraic over $K$ if $\alpha$ is the root of some monic polynomial with coefficients in $K$.

Show that if $\alpha$ and $\beta$ in $L$ are algebraic over $K$, then $\alpha \beta$ is algebraic over $K$.

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\item{2.} Let $w$ be a primitive cube root of unity.  Let $R={\bold{Z}}[w]$.  Let $\lambda=1-w$.  Show that $R/\lambda R \cong {\bold{Z}}/3{\bold{Z}}$.

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\item{3.} Let $G$ be the group of $2\times 2$ invertible matrices of determinant 1 with coefficients in the field of 3 elements.

(a) Show that $G$ has order 24.

(b) Find the number of 3-Sylow subgroups of $G$.

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\item{4.} Let $G$ be a finite $p$-group, $p$ prime, $V$ a finite dimensional vector space over the field ${\bold{F}}_p$ of $p$ elements.  Suppose $G$ acts linearly on $V$ (i.e. there is a homomorphism from $G$ into the group $GL(V)$ of invertible linear transformations from $V$ to $V$).  Prove that $G$ has a non-zero fixed point:  that is, there is some $\alpha \neq 0$ in $V$ so that $\sigma(\alpha)=\alpha$ for all $\sigma$ in $G$.

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\item{5.} Let $L/K$ be a Galois extension of fields with Galois group $G$.  Let $L=K[\alpha]$.  Define $tr(\alpha) = \dsize\sum_{r\in G} \sigma(\alpha)$.  Let $T_\alpha: L\to L$ be the $K$-linear transformation defined by $T_\alpha(\beta)=\alpha \beta$.  Show that $tr(\alpha)$ is the trace of the linear transformation $T_\alpha$.

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\item{6.} Prove that for any prime $p$, there are at least four isomorphism classes of groups of order $p^3$.

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\item{7.} A ${\bold{Z}}$-module $M$ is flat if for any short exact sequence $0\to A\to B\to C\to 0$ of ${\bold{Z}}$-modules, the sequence $0\to M\otimes A\to M\otimes B \to M\otimes C \to 0$ is exact.

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(a) State and prove a criterion for flatness as follows:  $M$ is flat if and only if for any 

\hskip15pt homomorphism $f: E\to F$ of ${\bold{Z}}$-modules, if $t$ is {\hbox to .25in{\hrulefill}}jective, then $M\otimes f$ is 

\hskip15pt {\hbox to .25in{\hrulefill}}jective.

(b) Give an example of a non-flat ${\bold{Z}}$-module.

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\item{8.} Let $K$ be a field, $M$ a $K$-vector space.  Let $M^*$ = $\text{Hom}_R(M,K)$.  Show that the canonical map $M\to M^{**}$ is surjective if and only if $M$ is finite dimensional.



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