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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 9\\
Due: April 11 (Monday)

\section{Exercises}

\begin{enumerate}[(1)]

\bigskip

\item Let $a,b$ be positive integers. If $p$ is a prime, then show that at least one of $a,b,ab$ is a square in $\bF_p$. In particular, the polynomial 
\[
(x^2-2)(x^2-3)(x^2-6) \in \bZ[x]
\]
has a root modulo $p$ for all primes $p$, but no root in $\bQ$.

\item
  \begin{enumerate}
  \item Let $G$ be a finite group. Let $X, Y \subseteq G$ be subsets such that $|X| + |Y| > |G|$. Show that every element of $G$ can be written as $xy$ where $x \in X$ and $y \in Y$.
  \item Let $K$ be a finite field. Use (a) to show that every element in $K$ is a sum of two squares, i.e., for all $a \in K$ there exists $x,y \in K$ such that $a = x^2 + y^2$. 
  \end{enumerate}

\item Describe $\bF_4$ and $\bF_8$ {\it explicitly}. More specifically, find a way to list its elements and to describe addition, multiplication, division. Using your description, find a generator for its multiplicative group (the nonzero elements under multiplication).

\item Calculate the Galois group of $(x^2-2)(x^2-3)(x^2-5)$ over $\bQ$. What are all of the subfields between $\bQ$ and its splitting field?

\item Let $k$ be any field and $k(t)$ its function field. Consider the automorphism $\sigma \colon k(t) \to k(t)$ defined by $\sigma(f(t)) = f(t+1)$. Show that the fixed subfield of $\sigma$ is $k$ if $k$ has characteristic $0$, and is $k(t^p-t)$ if $k$ has characteristic $p>0$.

\end{enumerate}

\section{Further reading}

When you compute Galois groups of low degree polynomials, you should expect to see some small finite groups. So it might be a good idea to memorize the structure (subgroup lattice and which subgroups are normal) of the ``small'' groups, where small has different meanings, but for sure of size $\le 11$. For size 12, there are 5 groups up to isomorphism; there are many references for this, here's one:

\url{http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/group12.pdf}

The Galois group of the splitting field of a separable polynomial of degree $d$ is some subgroup of the symmetric group $S_d$. So again, it might be useful to familiarize yourself with the subgroups of $S_d$ when $d$ is small. In this case, $d \le 5$ is reasonable. Here's a table for the different subgroups of $S_5$ up to isomorphism:
\url{http://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S5#Table_classifying_isomorphism_types_of_subgroups}

\end{document}