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\begin{document}

\qtitle{Exam 2 -- March 27, 2020}
\qheader

Time: 2 hours. All problems will be weighted equally. You may make use
of all theorems proven in text if you cite them (correctly).

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\begin{problem}
  Which of the following polynomials are irreducible in the given
  ring? Explain, and if the polynomial is reducible then give an
  irreducible factorization.
  \begin{enumerate}
  \item $X^5+2X^2+X+1\in\bZ[X]$ (hint: reduce).
  \item $X^3-X^2-X-2\in\bQ[X]$.
  \item $X^4+2\in\bF_7[X]$
  \item $X^4+(2+i)X^2+(1-2i)X+5\in\bQ[i][X]$.
  \end{enumerate}
\end{problem}
\pagebreak

\begin{problem}
  Suppose $R$ is a ring with identity, $M$ is a (left) $R$-module,
  $N\subseteq M$. Denote by $p:M\to M/N$ the canonical
  homomorphism, and by $i:N\to M$ the (evident) inclusion
  homomorphism. 
  \begin{enumerate}
  \item Show that if there is an $R$-module homomorphism $s:M/N\to M$
    such that $p\circ s=id_{M/N}$ then $M\simeq N\oplus M/N$.
  \item Show that if there is an $R$-module homomorphism $s:M\to N$
    such that $s\circ i=id_N$ then $M\simeq N\oplus M/N$.
  \item Give an example of a ring $R$ with identity, an $R$-module $M$
    and a submodule $N$ such that $M\not\simeq N\oplus M/N$.
  \end{enumerate}
\end{problem}
\pagebreak

\begin{problem}
  Suppose $K$ is a field, $m$ and $n$ are relatively prime
  positive integers and $R=K[X,Y]/(X^m-Y^n)$. Show that $R$ is an
  integral domain.
\end{problem}
\pagebreak

\begin{problem}
  Suppose $K$ is a field, $V$ is a vector space of finite dimension,
  and $W$, $W'\subseteq V$ are subspaces of $V$. (i) Show that
  \begin{displaymath}
    \dim(W+W')=\dim(W)+\dim(W')-\dim(W\cap W').
  \end{displaymath}
  (ii) Use this (or any other argument you like) to show that
  $V\simeq W\oplus W'$ if and only if $W\cap W'=0$ and
  $\dim(W)+\dim(W')=\dim(V)$.
\end{problem}

\end{document}

