%%% Document Class

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


%%% Packages

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


%%% Variables

\newcommand{\courseName}{Math 480A2}
\newcommand{\hwNumber}{2}
\newcommand{\hwDueDate}{September 8, 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}

\textbf{Problem 1.} (3 points) Describe the elements which are units in the ring $R = \ZZ / 12 \ZZ$.  List the ideals of $R$ (express each as a set), and explain why each set is an ideal, and why every ideal is accounted for.

\vspace{3mm}

\textbf{Problem 2.} (2 points) Let $R, R'$ be commutative rings, let $\varphi: R \to R'$ be a homomorphism, and let $I'$ be an ideal in $R'$.  Prove that the preimage
\[
	\varphi^{-1}(I') = \{r \in R : \varphi(r) \in I' \}
\]
is an ideal in $R$.

\vspace{3mm}

\textbf{Problem 3.} (2 points) Let $f(x) = 3x^5 + x^4 + 5x^3 + 4x^2 + 4x + 6$ and let $g(x) = x^3 + x + 1$ be polynomials in $\ZZ[x]$.  Use polynomial long division to find quotient and remainder polynomials $q$ and $r$ such that $f(x) = q(x) g(x) + r(x)$, where $r$ has degree less than the degree of $g$.

\vspace{3mm}

\textbf{Problem 4.} (3 points) Let $R = \ZZ / 2 \ZZ$, and let $F = R[x] / (x^2 + x + 1)$, where $x^2 + x + 1$ is interpreted as a polynomial with coefficients in $\ZZ / 2 \ZZ$.  How many elements does $F$ have?  (Hint: use polynomial long division to express each element of $F$ in a simple ``canonical'' form.)  Write out the addition and multiplication table of $F$, and explain why $F$ is a field.

\end{document}