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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 30 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}

\noindent Unless otherwise specified, assume that $R$ is a commutative
ring with $1$ and $F$ is a field. \medskip

\begin{enumerate}

\item For each of the following ideals $I$ of $\Z[x]$, determine
  whether they are principal, and then find $I\cap\Z$ and $\Z[x]/I$,
  up to isomorphism.
  \begin{enumerate}
  \item $I=(x^2+1,\;x^2+x+3)$,
  \item $I=(2,\;x^2+x+1)$. \\
  \end{enumerate}



\item Fix $a\in\Z$ and let $\phi_a:\Z[x]\to\Z$ be the evaluation map,
  $\phi_a(f(x))=f(a)$.
  \begin{enumerate}
  \item If $I=(3,x^2+1)$ in $\Z[x]$, show that $\phi_a(I)=\Z$.
  \item Show by example how Part~(a) can fail if $3$ is replaced with
    a different odd prime $p$. \\
  \end{enumerate}



\item 
  \begin{enumerate}
  \item Show that an ideal $I=(f)$ of $F[x]$ is maximal if and only if
      $f$ is irreducible.
  \item Show by example how Part~(a) can fail in $R[x]$, where $R$ is
    not a field.
  \item Consider the ideal $I=(x_1-a_1,\dots,x_n-a_n)$ in
    $R[x_1,\dots,x_n]$, for distinct $a_1,\dots,a_n\in R$. Formulate
    and prove necessary and sufficient conditions on $R$ that will
    ensure that $I$ is a maximal ideal. \\
  \end{enumerate}



\item For $f(x)\in R[x]$, substituting $a\in R$ for $x$ determines a
  \emph{polynomial function} $f\colon R\to R$, where $a\mapsto f(a)$.
\begin{enumerate}
\item If $R$ is an infinite integral domain, show that the mapping
  $f(x)\mapsto f$ assiging to each polynomial in $R[x]$ to its
  corresponding polynomial function is 1--1.
\item If $R$ is finite, show that there must exist distinct
  polynomials $f(x)$ and $g(x)$ in $R[x]$ with identical associated
  functions $f$ and $g$, i.e., $f(a)=g(a)$ for all $a\in R$. \\
\end{enumerate}



\item Let $\F_q$ be the finite field of order $q=p^n$. 
  \begin{enumerate}
    \item Show that every function $f\colon\F_q\to\F_q$ can be written
      uniquely as a polynomial in the quotient $\F_q[x]/I$, where
      $I=(x^q-x)$. How many such functions are there? 
    \item Generalize Part~(a) to multivariate functions
      $f\colon\F_q^n\to\F_q$.
    \item A \emph{Boolean network} is an $n$-tuple $(f_1,\dots,f_n)$
      of functions $f_i\colon\F_2^n\to\F_2$. How many $6$-node Boolean
      networks are there? Express your answer in scientific notation. \\
  \end{enumerate}


  \end{enumerate}

\end{document}





