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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 10.1  \\ Fall 2019 \hfill Due November 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $K/F$ be fields and suppose $S, T \subset K$.  Prove that $F(S)(T) = F(S \cup T)$.
\end{exercise}

\begin{exercise}\label{prev}
Let $K/F$ be fields.  Suppose that $\alpha \in K$ is one root of an irreducible quadratic polynomial $f(X) \in F[X]$.  Prove that $F(\alpha) = \{a + b \alpha \, | \, a, b \in F \}$.
\end{exercise}

\begin{exercise}
Let $p, q \in \Z$ be distinct prime numbers. 
\begin{itemize}
\item[\bf a.] Prove that $X^2 - q$ is irreducible over $\Q(\sqrt{p}) = \{ a+ b \sqrt{p}  \, | \, a , b \in \Q\}$.  

\item[\bf b.] Use part {\bf a} and Exercise \ref{prev} to conclude that
\[
\Q(\sqrt{p}, \sqrt{q}) = \{ a + b \sqrt{p} + c \sqrt{q} + d \sqrt{pq} \, | \, a, b, c, d \in \Q \}.
\]

\item[\bf c.] Show that $\Q(\sqrt{p}, \sqrt{q}) = \Q(\sqrt{p} + \sqrt{q})$.

\end{itemize}
\end{exercise}

\begin{exercise}\label{irred}
Let $F$ be a field.  The quotient field of the polynomial ring $F[X]$ is denoted $F(X)$ and is called the {\em field of rational functions} in $X$.  Let $n \in \N$ consider the polynomial $f(X,T) = T^n - X \in F[X,T]$.

\begin{itemize}

\item[\bf a.]  Show that $f$ is irreducible as a polynomial in $X$ over the field $F(T)$ of rational functions in $T$.  

\item[\bf b.]  Use (a corollary to) Gauss' lemma to conclude that $f$ is irreducible in $F[X,T]$.

\item[\bf c.]  Use similar reasoning to conclude that $f$ is irreducible as a polynomial in $T$ over the field $F(X)$.

\item[\bf d.]  Conclude that for $n \ge 2$, $X^{1/n}$ is ``irrational,'' i.e.\ $X^{1/n} \not \in F(X)$.  

\end{itemize}

\end{exercise}

\begin{exercise}
Let $p \in \Z$ be a prime number and let $\mathbb{F}_p = \Z/p\Z$ be the field with $p$ elements.  Show that the extension $\mathbb{F}_p(X^{1/p})$ of $\mathbb{F}_p(X)$ contains every root of $T^p - X \in \mathbb{F}_p(X)[T]$.  
\end{exercise}

\end{document}
