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\noindent
\parbox{2.25in}{\bf Math 4720/5210}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Oct. $27^\text{th}$, 2021}

\vspace{0.2in}

\begin{description}
\item[\#1 Easy Euclid] Let $E$ be a Euclidean domain equipped with norm $\delta:E-\{0\} \to \mathbb{Z}_{\geq 0}$. Suppose $a,b\in E$ are non-zero are non-zero associates. Show that $\delta(a)=\delta(b)$. Is the converse true?

\item[\#2 Euclid's Revenge!] A quotient of $\mathbb{Q}[x]$.
   \begin{enumerate}[(a)]
   \item Find the GCD of $x^3-2x^2+1$ and $x^2-x+3$ in $\mathbb{Q}[x]$ and express it as a linear combination (i.e. run the Extended Euclidean Algorithm). 
   \item Let $I=(x^2-x+3)$. Is $x^3-2x^2+1+I$ zero, a zero divisor, or a unit in $\diagquotient{\mathbb{Q}[x]}{I}$? Prove your result (If zero, why? If a zero divisor, what is a non-zero element that multiplied by gives zero? If a unit, what's its inverse?).
   \item Let $I=(x^3-8)$. Is $x^2-4+I$ zero, a zero divisor, or a unit in $\diagquotient{\mathbb{Q}[x]}{I}$? Prove your result (If zero, why? If a zero divisor, what is a non-zero element that multiplied by gives zero? If a unit, what's its inverse?).
   \end{enumerate}

\item[\#3 A Rational Problem] As in the Factorization Handout, compute the inverse of $x^2+x+2+I$ in $\diagquotient{\mathbb{Q}[x]}{I}$ where $I=(x^3-3)$. Then use this result to rationalize the fraction $\dfrac{1}{2+3^{1/3}+3^{2/3}}$ (i.e. write this fraction as $a+b \cdot 3^{1/3}+c \cdot 3^{2/3}$ for some $a,b,c \in \mathbb{Q}$).


\item[\#4 Prime, maximal, both, or neither?] Identify the following ideals as prime, maximal, both, or neither.

\begin{enumerate}[(a)]
\item $(x^2-5)$ in $\mathbb{Q}[x]$ 
\item $(x^2-5)$ in $\mathbb{R}[x]$ 
\item $(x^2+1)$ in $\mathbb{Q}[x]$ 
\item $(x^2+1)$ in $\mathbb{Z}[x]$
\end{enumerate}


\end{description}

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