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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 9.2  \\ Fall 2019 \hfill Due November 13}}}

\large

\vspace{1cm}

\begin{exercise}
Let $F$ be a field and define
\[
F[X ; \Q^+_0] = \left\{ \sum_{r \in \Q_0^+} a_r X^r \, \bigg| \, a_r \in F, \text{ almost all } a_r = 0 \right\},
\]
where ``almost all'' means all but finitely many.  

\begin{itemize}

\item[\bf a.]  Let $r \in \Q^+$.  Show that the rule $f(X) \mapsto f(X^r)$ defines an automorphism $E_r$ of $F[X; \Q^+_0]$.

\item[\bf b.]  Given $f_1, \ldots, f_n \in F[X ; \Q_0^+]$, show that there is an $m \in \N$ so that $E_m(f_i) \in F[X]$ for all $i$. 

\item[\bf c.] Show that $F[X; \Q^+_0]$ is a GCD domain.

\end{itemize}

\end{exercise}


\begin{exercise}

Fix $r_0 \in \Q^+$ and let $S = \{X^{r} \, | \, r>r_0 \} \subset F[X;\Q_0^+]$.

\begin{itemize}

\item[\bf a.]  Show that $(S)$ cannot be finitely generated.

\item[\bf b.]  Find (with proof) $\gcd S$.

\end{itemize}

\end{exercise}

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