%%% Template for MA 361 homework, after Eric J. Malm's hmcpset class
%
%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\usepackage{tikz}
\usetikzlibrary{arrows}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
%
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Inn}{\operatorname{Inn}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{4}  % the homework number
\newcommand\DUEDATE{2/14/2020} % the due date
\newcommand\COURSE{MA 362}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
\textbf{The rational root test}
\begin{enumerate}
\item Let
\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in \ZZ[x] \qquad (a_n \neq 0), \]
and suppose $p(\alpha) = 0$ for some $\alpha \in \QQ$. Prove that if $\alpha = \frac{r}{s}$ is in lowest terms (i.e. $r, s \in \ZZ$ are relatively prime), then $r$ divides $a_0$ and $s$ divides $a_n$.

\item Use part (a) to quickly find a rational root of $9x^5 + 8x^3 + 2x^2 - x + 2$.

\item Use part (a) to show that $2x^3 - 7x + 3$ is irreducible in $\QQ[x]$.
\end{enumerate}


\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[2]
Let $p$ be a prime number. Show that the polynomial
\[ x^n - p \]
is irreducible over $\QQ$ for any integer $n \geq 1$.
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[3]
Show that the polynomial
\[ (x-1)(x-2) \cdots (x-n) - 1 \]
is irreducible over $\QQ$ for any integer $n \geq 1$. [\textsc{hint:} Suppose it factors as $f(x)g(x)$ and consider the polynomial $f(x) + g(x)$ at the values $x = 1, \ldots, n$.]
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[4]
Let $n$ be a positive integer. Prove that the polynomial
\[ 1 + x + x^2 + \cdots + x^{n-1} \]
is irreducible in $\QQ[x]$ if and only if $n$ is prime.
\end{problem}

%\begin{proof}
%
%\end{proof}



\end{document}
