\documentclass[12pt]{amsart}

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsxtra}
\usepackage{mathrsfs}

\setlength{\hfuzz}{4pt}

\newenvironment{problab}[1]
{\noindent\textbf{Problem #1}.}
{\vskip 6pt}

\theoremstyle{remark}
\newtheorem*{solu}{Solution}

\newenvironment{enumalph}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\alph{enumi})}}}
{\end{enumerate}}

\newenvironment{enumroman}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}}
{\end{enumerate}}

\newcommand{\C}{\mathbb C}
\newcommand{\F}{\mathbb F}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}

\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\img}{img}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\opchar}{char}

\usepackage{fullpage}

\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#5}
\date{4 February 2015; due Wednesday, 11 February 2015}

\maketitle

\begin{problab}{5.1}
Is the field of constructible numbers Galois over $\Q$?  Give a proof or a counterexample.
\end{problab}

\begin{problab}{5.2}
Let $a \in \Q$, and let $f(X)=X^3+aX^2-(a+3)X+1 \in \Q[X]$ be irreducible.
\begin{enumalph}
\item Show that $f$ has Galois group $\Z/3\Z$.
\item Let $F=\Q(\alpha)$ with $\alpha$ a root of $f$.  Show that the map $\alpha \mapsto 1/(1-\alpha)$ generates $\Gal(F/\Q)$.
\end{enumalph}
\end{problab}

\begin{problab}{5.3}
Let $D(f)$ be the discriminant of a polynomial $f \in F[X]$.  
\begin{enumalph}
\item Show that 
\[ D(X^n-1)=(-1)^{\binom{n}{2}+n-1}n^n. \]
\item Let $h \in F[X]$ be monic and let $g(X)=(X-a)h(X)$ with $a \in F$.  Show that
\[ D(g)=h(a)^2 D(h). \]
\item Show that 
\[ D(X^{n-1}+X^{n-2}+\dots+1)=(-1)^{(n-1)(n+2)/2}n^{n-2}. \]
\end{enumalph}
\end{problab}


\end{document}