%%% Document Class

\documentclass[11pt]{letter}
\usepackage[margin=1in]{geometry}


%%% Packages

\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{mathtools}


%%% Variables

\newcommand{\courseName}{Math 480A2}
\newcommand{\hwNumber}{4}
\newcommand{\hwDueDate}{September 22, 2022}


%%% Definitions

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\renewcommand{\implies}{\rightarrow}
\renewcommand{\iff}{\leftrightarrow}

\renewcommand{\vec}[1]{\mathbf{#1}}


%%% Document

\begin{document}

\begin{center}
	{\large \courseName, Homework \hwNumber}\\
	Due \hwDueDate\\
\end{center}

{\small \textit{Homework is graded out of a total of 10 points.  Collaboration is permitted, but you must list all coauthors on a problem's solution at the top of the page, and your writing must be your own.}}

\vspace{5mm}

For the following problems, let $F_2 = \ZZ / 2 \ZZ$ denote the binary field.

\vspace{2mm}

\textbf{Problem 1.} (3 points) Find the factorization of $x^{16} - x$ into irreducible factors over $F_2$.  (\textit{Hint:} You may make use of solutions from previous homework assignments.)

\vspace{3mm}

\textbf{Problem 2.} (2 points) Describe a finite field $K$ of order 16 over $F_2$, presented as the quotient of $F_2[x]$ by an irreducible polynomial.  Describe the general form of an element of $K$ as a linear combination of small powers of $\bar{x}$ in the quotient, and give the relation satisfied by $\bar{x}^4$.

\vspace{3mm}

\textbf{Problem 3.} (3 points) Find a multiplicative generator of $K^\times$ by identifying a nonzero element $\alpha \in K$ satisfying $\alpha^3 \neq 1$ and $\alpha^5 \neq 1$.

\vspace{4mm}

\textbf{Problem 4.} (2 points) Let $\beta = \alpha^5$, where $\alpha$ is the multiplicative generator found in the last problem.  Show that the set $K' = \{0, 1, \beta, \beta^2\}$ is a subfield of $K$ of order 4 by writing out the addition and multiplication tables of these elements.

\textbf{Challenge.} (1 bonus point) Describe an explicit isomorphism between $K'$ and $F_2[y] / (y^2 + y + 1)$.

\end{document}