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\pagestyle{fancy}
\lhead{MATH 522--01}
\chead{Problem Set \#5\answer{ solutions}}
\rhead{}% Your name here!
\lfoot{}
\cfoot{\answer{Page \thepage\ of \pageref{LastPage}}}
\rfoot{due Thursday, April 9, 2015}

\begin{document}
\begin{enumerate}
\item \question{If $F$ has characteristic $p$ and $a\in F$,consider
    the polynomial $f(x)=x^p-a\in F[x]$. Prove that $f(x)$ is either
    irreducible over $F$ or $f(x)$ splits in $F$.}

  \answer{}

\item \question{Let $F=\mathbb Q(\sqrt 2,\sqrt 3)$. Find a polynomial
    $f(x)\in\mathbb Q[x]$ such that $F\cong\mathbb Q[x]/\langle
    f(x)\rangle$, and find a basis for $F$ considered as a vector
    field over $\mathbb Q$.}

  \answer{}

\item \question{Let $K$ be an extension of field $F$. If $\gamma\in K$
    and $a,b\in F$ with $a$ nonzero, prove that
    $F(a\gamma+b)=F(\gamma)$ (note: this is set-equality, not simply
    isomorphism).}

  \answer{}

\item \question{Let $\alpha\in\mathbb R$. Prove that $\mathbb
    Q(\alpha)\cong\mathbb Q(x)$ if and only if $\alpha$ is
    transcendental, i.e., if $\alpha$ is not the root of any
    polynomial with rational coefficients.}

  \answer{}

\item \question{Prove that there is no irreducible polynomial in
    $\mathbb Q[x]$ which is zero at both $x=\sqrt 5$ and $x=\sqrt 7$.}

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