%%% 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{5}  % the homework number
\newcommand\DUEDATE{2/28/2020} % 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]
Prove that $\QQ(\sqrt{5}, \sqrt{7}) = \QQ(\sqrt{5} + \sqrt{7})$.
\end{problem}

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


\newpage


\begin{problem}[2]
The following complex numbers are algebraic over $\QQ$. Find the minimal polynomial for each over $\QQ$.
\begin{enumerate}
\item $\sqrt{3} + \sqrt{2}$
\item $\sqrt{3 + i\sqrt{2}}$
\item $\sqrt{3 + 2\sqrt{2}}$
\end{enumerate}
\end{problem}

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


\newpage


\begin{problem}[3]
Let $E$ be an extension field of a finite field $F$, where $F$ has $q$ elements. Let $\alpha \in E$ be algebraic over $F$ of degree $n$. Prove that $F(\alpha)$ has $q^n$ elements.
\end{problem}

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


\newpage


\begin{problem}[4]
Let $\alpha = e^{2\pi i/5} = \cos(2\pi/5) + i \sin(2\pi/5) \in \CC$.
\begin{enumerate}
\item Find the minimal polynomial of $\alpha$ over $\QQ$.
\item For which positive integer $k$ is $\{1, \alpha, \alpha^2, \ldots, \alpha^k\}$ a basis for $\QQ(\alpha)$ as a vector space over $\QQ$? Express $\alpha^{-1} = e^{-2\pi i/5}$ as a $\QQ$-linear combination of basis elements.
\item Find the minimal polynomial for $2\cos(2\pi/5)$ over $\QQ$. [\textsc{hint:} Consider $\alpha + \alpha^{-1}$].
\item Use the polynomial from part (c) to find a simple expression for $\cos(2\pi/5)$.
\end{enumerate}
\end{problem}

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



\end{document}
