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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Abstract Algebra Section 
\endtitle
\author
\bf Fall 1995
\endauthor
\endtopmatter
\document

\vglue\baselineskip

\t{Part I}{Do three (3) of these problems.}

\s{I.1}
{Show that the alternating group $A_6$ has no subgroup of prime index.}

\s{I.2}
{Let $R$ be a commutative domain in which every element $x$ satisfies
  $x^n=x$ for some $n>1$ (depending on $x$). Show that $R$ is a field of
  positive characteristic. Is $R$ necessarily finite?}

\s{I.3}
{Let $V$ be a finite-dimensional vector space and let $\phi: V\to V$
  be an endomorphism. Suppose that, for some $v\in V$ and $k\ge 1$,
  $\phi^k(v)=0$ but $\phi^{k-1}(v)\neq 0$. Prove:
\roster
  \item The subspace $W$ of $V$ that is generated
  by $\{v,\phi(v),\ldots,\phi^{k-1}(v)\}$ is $\phi$-invariant (i.e.,
  $\phi(W)\subseteq W$) and satisfies $\dim(W)=k$.
  \item The minimal polynomial $m(X)$ of $\phi$ is divisible by $X^k$.
\endroster}

\s{I.4}
{Let $F$ be a field of characteristic $p>0$ and let $f(X)\in F[X]$
  be an irreducible polynomial which is not separable (i.e., $f(X)$ has
  repeated roots). Show that $f(X)=g(X^p)$ for some irreducible polynomial
  $g(X)\in F[X]$.}

\vglue\baselineskip
\t{Part II}{Do two (2) of these problems.}

\s{II.1}
{Show that there is no simple group of order 56 (without quoting
  Burnside's $p^aq^b$-Theorem or special cases thereof). }

\s{II.2}
{Let $M\neq 0$ be a finitely generated torsion module over a
  commutative PID $R$.
  \roster
  \item Show that $M$ is {\it indecomposable\/} (i.e., $M$ is not the direct
    sum of two nonzero submodules) if and only if $M\cong R/p^nR$ for some
    irreducible element $p$ of $R$ and some $n>0$.
  \item Show that $M$ is {\it irreducible\/} (i.e., $M$ has no submodules
    other than $0$ and $M$) if and only if $M\cong R/pR$
    for some irreducible element $p$ of $R$.
  \endroster }

\s{II.3}
{Let $F$ be a finite field and $n$ a positive integer. Prove that there
  exists an irreducible polynomial over $F$ of degree $n$.}

\enddocument