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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 11\\
Due: April 22

\section{Exercises}

\begin{enumerate}[(1)]

\item A field $k$ is {\bf perfect} if either $k$ has characteristic $0$, or $k$ has characteristic $p$, and the equation $x^p - a = 0$ has a solution for all $a \in k$.
  \begin{enumerate}
  \item Show that finite fields and algebraically closed fields are perfect.
  \item Show that $k$ is perfect if and only if every algebraic extension of $k$ is separable.
  \end{enumerate}

\item 
\begin{enumerate}
\item Let $E$ be a field of characteristic different from $2$. Let $F = E(\sqrt{D})$ be a quadratic extension (so $D \in E$ but $\sqrt{D} \notin E$). Show that ${\rm N}_E^F(a + b\sqrt{D}) = a^2 - Db^2$ and that ${\rm Tr}_E^F(a + b\sqrt{D}) = 2a$.
\item Apply Hilbert's theorem 90 to the extension $\bQ(i)/\bQ$ to show that all solutions to $a^2 + b^2 = 1$ with $a,b \in \bQ$ are of the form 
\[
a = \frac{s^2 - t^2}{s^2 + t^2}, \qquad b = \frac{2st}{s^2+t^2}
\]
for some $s,t \in \bQ$.
\item Let $D$ be an integer such that $\sqrt{D}$ is not an integer. Generalize (b) to determine the solutions to $a^2 - Db^2 = 1$ for $a,b \in \bQ$.
\end{enumerate}

\item Read Example 7 from Lang, \S VI.2. Use it to show that the Galois groups of these two polynomials (over $\bQ$) is the full symmetric group $S_4$:
\[
x^4 + 2x^2 + x + 3, \qquad x^4 + 3x^3 - 3x - 2.
\]

\item Let $K/k$ be a finite separable extension of degree $n$. Write $K=k(\theta)$ where $\theta \in K$ and let $\theta_1, \dots, \theta_n$ be the conjugates of $\theta$ in $\ol{k}$. Define an equivalence relation $\theta_i \sim \theta_j$ if $k(\theta_i) = k(\theta_j)$ as subfields of $\ol{k}$. 
\begin{enumerate}
\item Show that the equivalence classes all have the same size.
\item Now assume $n$ is prime. Conclude that either all of the subfields are the same or are all different. In particular, show that if $K$ contains two of the $\theta_i$, then in fact $K/k$ is a cyclic extension.
\end{enumerate}
\end{enumerate}

\section{Further reading}

I've chosen not to put any complicated Galois group calculations on the homework. If you'd like to do some, there are plenty of resources for that. For example, you might read Section 14.8 of Dummit, Foote, {\it Abstract Algbera} (third edition) and look at its exercises.

\end{document}
