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\usepackage{enumerate}
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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 10\\
Due: April 15

\section{Exercises}

\begin{enumerate}[(1)]

\item Consider each polynomial below as belonging to $\bQ[x]$, and determine the Galois group of its splitting field $K$ over $\bQ$. 
  \begin{enumerate}
  \item $x^3 + x + 1$
  \item $x^3 - 3x + 1$
  \item $x^3-2x+1$
  \item $x^3-x+1$
  \end{enumerate}

\item
  \begin{enumerate}
  \item Let $\phi(n)$ be the Euler totient function, i.e., the number of positive integers $\le n$ which are relatively prime to $n$. Show that for each integer $m$, there are only finitely many $n$ such that $\phi(n)=m$.
  \item Let $k$ be a finite extension of $\bQ$. Show that $k$ contains only finitely many roots of unity.
  \end{enumerate}

\item Let $\ell, p$ be prime numbers. This exercise describes how the cyclotomic polynomial $\Phi_\ell(x) = (x^\ell - 1) / (x -1)$ factors in $\bF_p[x]$.
  \begin{enumerate}
  \item If $p=\ell$, show that $\Phi_\ell(x) = (x-1)^{\ell-1}$.
  \item If $p\ne \ell$, let $\zeta$ be a primitive $\ell$th root of unity in $\overline{\bF}_p$. Show that $p^n = 1 \pmod \ell$ if and only if $\zeta \in \bF_{p^n}$. Conclude that the degree $d$ of the minimal polynomial of $\zeta$ over $\bF_p$ is the order of $p$ in $\bF_\ell^\times$. 

Conclude that $\Phi_\ell(x)$ factors into $(\ell-1)/d$ distinct irreducible polynomials of degree $d$.
  \end{enumerate}

\item Let's continue with Example 3 in Lang, \S VI.2. Which order $8$ group is the Galois group $G$? Draw the diagram of subfields of $K$. (You can draw this by hand if you like, though it's good practice to learn how to type this. For example, you can use {\tt xypic} to do this.)

\end{enumerate}

\end{document}
