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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 11.1  \\ Fall 2019 \hfill Due December 2}}}

\large

\vspace{1cm}

\begin{exercise}
Let $F$ be a field and suppose $f \in F[X]$ has positive degree $n$.  Use the division algorithm to show that the cosets of $1$, $X$, $X^2, \ldots, X^{n-1}$ in $F[X]/(f)$ form an $F$-basis for $F[X]/(f)$.
\end{exercise}

\begin{exercise}
Let $K/F$ be fields and suppose $t \in K$ is transcendental over $F$.  Show that the evaluation map $E_{t} : F[X] \to K$ yields an isomorphism $F(X) \cong F(t)$.  
\end{exercise}

\begin{exercise}
Let $F$ be a field and let $a \in F$.  If $a \ne \Box$ in $F$, prove that $[F(\sqrt{a}): F] = 2$.
\end{exercise}

\begin{exercise}
Let $p$, $q$ be distinct primes.  Find the minimal polynomial of $\sqrt{p} + \sqrt{q}$ over $\Q$.  [{\em Suggestion:}  Use Exercises 3 and 10.1.3c.]
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