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\begin{document}
\title{Homework $\#$9}
\author{Not to be turned in}
\maketitle
\subsection*{Problem 1} Find all ideals of $\mathbb{Q}[x]$ containing $(x^4-1)$. For each ideal you find, determine whether it is maximal, prime, both, or neither.

\subsection*{Problem 2 (D$\&$F 9.2.4)} Let $F$ be a finite field. Prove that $F[x]$ contains infinitely many primes.

\subsection*{Problem 3 (D$\&$F 9.4.1,2)}
Determine whether the following polynomials are irreducible in the rings indicated. For those that are reducible, determine their factorization into irreducibles. Let $\mathbb{F}_p$ denote the finite field $\mathbb{Z}/p\mathbb{Z}$:
\begin{itemize}
\item[(a)] $x^2+x+1$ in $\mathbb{F}_2[x]$.
\item[(b)] $x^4-4x^3+6$ in $\mathbb{Z}[x]$.
\item[(c)] $x^6+30x^5-15x^3+6x-120$ in $\mathbb{Z}[x]$.
\item[(d)] $x^4+1$ in $\mathbb{F}_5[x]$.
\item[(e)] $x^3+x+1$ in $\mathbb{F}_3[x]$.
\item[(f)] $x^4+4x^3+6x^2+2x+1$ in $\mathbb{Z}[x]$.
\end{itemize}

\subsection*{Problem 4 (D$\&$F 9.4.3)} Show that the polynomial $(x-1)(x-2)\cdots(x-n)-1$ is irreducible in $\mathbb{Z}[x]$ for all $n\geq 1$.

\noindent [Hint: Note that in a UFD, the fundamental theorem of arithmetic holds.]


\subsection*{Problem 5 (D$\&$F 9.4.6)} Construct a field of order $9$. You may exhibit this field as $F[x]/(f(x))$ for some $F$ and $f$. You may use D\&F problem 9.2.2.

\subsection*{Problem 6 (D$\&$F 9.4.11)} 
Prove that $x^2+y^2-1$ is irreducible in $\mathbb{Q}[x,y]$. 

\subsection*{Problem 7 (D$\&$F 9.4.14)} Factor the polynomials $x^8-1$ and $x^6-1$ into irreducibles in each of the following rings:
\begin{itemize}
\item[(a)] $\mathbb{Z}[x]$
\item[(b)] $\mathbb{Z}/2\mathbb{Z}[x]$
\item[(c)]$\mathbb{Z}/3\mathbb{Z}[x]$

\end{itemize} 

\end{document}