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\pagestyle{fancy}
\lhead{MATH 522--01}
\chead{Problem Set \#3\answer{ solutions}}
\rhead{}% Your name here!
\lfoot{}
\cfoot{\answer{Page \thepage\ of \pageref{LastPage}}}
\rfoot{due Thursday, February 26, 2015}

\begin{document}
\begin{enumerate}
\item \question{For a field $F$, let $F(x)$ denote the field of
    fractions of the ring $F[x]$; in other words, the field of
    rational functions with coefficients in $F$. Prove that there is
    no element $f\in F(x)$ such that $f^2=x$.}

\answer{}

\item \question{Prove that, if $I$ is a prime ideal of a ring $R$,
    then $I[x]$ is a prime ideal of $R[x]$.}

\answer{}

\item \question{Describe a field of order 25 and a field of order 27.}

\answer{}

\item \question{Let $\alpha\in\mathbb R$ be a \emph{transcendental
      number}, i.e., there is no nonzero polynomial with integer
    coefficients which has $\alpha$ as a zero. Prove that $\alpha^2$
    cannot be written as a linear combination of $\alpha$ and a
    rational number, i.e. $\alpha^2=\alpha x+y$ for $x,y\in\mathbb
    Q$.}

  \answer{}

\item \question{Prove that for a principal ideal domain $D$ with $p$
    an irreducible element of $D$, $D/\langle p\rangle$ is a
    field. Show that if $D$ is merely an integral domain, $D/\langle
    p\rangle$ may not be a field.}

  \answer{
  }
\end{enumerate}
\end{document}