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\begin{document}

\noindent Math 742, Spring 2016\\
Takehome exam 2\\
Due: May 13, 12pm (slide under my office door, 321 VV)

~

\begin{itemize}
\item Do not discuss this exam with anyone else.

\item You may use your notes and textbook and use as much time as you like up until the due date. You are free to use any results in the book that we covered and any previous exercises. 

\item No late exams will be accepted. If necessary, you can email me your solutions. 

\item This counts for 4 homework assignments and all problems will be graded. \end{itemize}

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $k$ be a field. Define $k^{\rm fra}(\!(t)\!)$ to be the set of formal expressions $\eta = \sum_{n \ge n_0} a_n t^{n/N}$ where $a_n \in k$, $n_0$ is an integer, and $N$ is a positive integer, which depends on $\eta$.\footnote{In other words, this is the set of power series where the exponents are allowed to be arbitrary rational numbers, provided that the exponents that appear are bounded from below and have bounded denominator.} You can add and multiply these expressions in the same way you add and multiply power series and it turns out that $k^{\rm fra}(\!(t)\!)$ is a field (you don't need to prove this).
  \begin{enumerate}
  \item Let $k(\!(t)\!)$ be the field of Laurent series, i.e., the subfield consisting of expressions such that the denominator $N$ must be $1$. Show that $k^{\rm fra}(\!(t)\!)$ is an algebraic extension of $k(\!(t)\!)$.
  \item A harder fact is that if $k$ is algebraically closed of characteristic $0$, then $k^{\rm fra}(\!(t)\!)$ is also algebraically closed (you don't need to prove this). But this is false in positive characteristic: Suppose $k = \ol{\bF}_p$. Show that $x^p - x - \frac{1}{t}$ has no solution in $k^{\rm fra}(\!(t)\!)$.
  \end{enumerate}

\item 
\begin{enumerate}
\item Let $E$ be a field of characteristic different from $2$. Suppose $F$ is a Galois extension of $E$ such that the Galois group is isomorphic to $\bZ/2 \times \bZ/2$. Show that there exist $x,y \in E$ such that $F = E(\sqrt{x}, \sqrt{y})$ such that none of $x, y, xy$ are squares in $E$.

\item Conversely, pick $x,y \in E$ such that none of $x,y,xy$ are squares in $E$. Show that $E(\sqrt{x}, \sqrt{y})/E$ is Galois with Galois group $\bZ/2 \times \bZ/2$.
\end{enumerate}

\item 
  \begin{enumerate}
  \item Let $G$ be a finite group. Show that there exists a Galois extension $F/E$ in every characteristic whose Galois group is isomorphic to $G$.
  \item Let $G$ be a finite abelian group. Show that there exists $n>0$ and a subfield $K \subset \bQ(\zeta_n)$ ($\zeta_n$ is a primitive $n$th root of unity) such that $K/\bQ$ is Galois with Galois group $G$.
  \end{enumerate}


\item Let $f(x) = x^4 + ax^2 + b \in \bQ[x]$ be an irreducible polynomial with splitting field $K$.
  \begin{enumerate}
  \item Show that ${\rm Gal}(K/F)$ is a subgroup of $D_4$ (the dihedral group of size $8$).
  \item Show that the Galois group is $D_4$ for $x^4 - 4x^2 - 1$.
  \end{enumerate}

\item Let $A$ be an alternating $n \times n$ matrix with entries in a field $K$ such that ${\rm image}(A) = \ker(A)$ (both thought of as subspaces of $K^n$). In particular, $n$ is even. Define a symmetric bilinear form $\langle, \rangle$ on $K^n$ by 
\[
\langle (v_1, \dots, v_n), (w_1, \dots, w_n) \rangle = \sum_{i=1}^n v_i w_i.
\]
\begin{enumerate}
\item Given a subspace $W \subseteq K^n$, define the subspace
\[
W^\perp = \{x \in K^n \mid \langle x, w \rangle = 0 \text{ for all $w \in W$}\}.
\]
Show that $\dim W + \dim W^\perp = n$.

\item Let $V$ be a subspace of $K^n$ of dimension $n/2$. Define a bilinear form $(,)$ on $V$ by 
\[
(v,w) = \langle v, Aw \rangle.
\]
Show that $(,)$ is an alternating form.

\item Show that if $\dim V = n/2$ and $V \cap \ker A = V \cap (\ker A)^\perp = 0$, then $(,)$ is symplectic (i.e., nondegenerate) on $V$. 

\item Show that such a $V$ exists and conclude that $n$ is divisible by $4$.
\end{enumerate}

\end{enumerate}

\end{document}
