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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 6.1  \\ Fall 2019 \hfill Due October 23}}}

\large

\vspace{1cm}

\begin{exercise}
Let $F$ be a finite field with $q$ elements.   Let $E : F[X] \to F^F$ be the map that sends each polynomial to the function it defines on $F$.  That is, $E(f)(a) = f(a)$ for all $a \in F$.  
\begin{itemize}

\item[\bf a.]  Show that every element of $F$ is a root of $\phi(X) = X^q - X \in F[X]$.  

\item[\bf b.]  Show that if $f(X) \in F[X]$ satisfies $f(a)=0$ for all $a \in F$, then $f(X) = g(X) \phi(X)$ for some $g(X) \in F[X]$.

\item[\bf c.] Prove that $E$ is a surjective homomorphism with kernel $(\phi(X))$, so that 
\[
F[X]/(\phi(X)) \cong F^F.
\]
[{\em Suggestions: } Use the fact that the pointwise evaluation maps $E_a$ are homomorphisms.  Surjectivity can be established directly, but it may be easier to use the pigeonhole principle.]  

\end{itemize}

\end{exercise}

\begin{exercise}
Let $R$ be a Euclidean domain with norm $N$.

\begin{itemize}

\item[\bf a.]  Let $a \in R$.  Prove that if $a \ne 0$ and $N(a)=0$, then $a \in R^{\times}$.

\item[\bf b.]  Prove that $N(1) \le N(a)$ for all nonzero $a \in R$.

\item[\bf c.]  Let $a \in R$, $a \ne 0$.  Show that $N(a) = N(1)$ if and only if $a \in R^{\times}$.  

\item[\bf d.]  Determine $\Z[i]^{\times}$.

\end{itemize}

\end{exercise}

\begin{exercise}
Find the quotient and remainder when $7+10i$ is divided by $2 - 3i$ in $\Z[i]$.  Show that there are four possible quotient-remainder pairs when $17$ is divided by $3-5i$.
\end{exercise}

\begin{exercise}
Let $\alpha = \frac{1+\sqrt{-3}}{2} \in \C$.  

\begin{itemize}

\item[\bf a.]  Show that $\alpha$ is a root of a monic quadratic polynomial in $\Z[X]$.

\item[\bf b.]  Prove that for every $z \in \C$ there is a $w \in \mathcal{P} = \{ s+t \alpha \, | \, s,t \in [0,1) \}$ so that $z-w \in \Z[\alpha]$.  That is, $\C = \mathcal{P} + \Z[\alpha]$.  [{\em Suggestion:}  Start by showing that $\{1 , \alpha \}$ is an $\R$-basis for $\C$.]

\item[\bf c.]  Exploit the symmetry of $\mathcal{P}$ to prove that $\Z[\alpha]$ with the norm $N(z) = |z|^2$ is a Euclidean domain.

\item[\bf d.]  Show that $\Z[\alpha]^{\times} = \{ \pm 1, \pm \alpha, \pm \overline{\alpha} \}$. 

\end{itemize}
 
\end{exercise}



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