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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 2 (due 25/9/2020)}
\begin{center}
\textbf{Some polynomial algebra}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] Let $R$ be a ring, $P\in R[x]$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $(x-y)$ divides $(x^{n}-y^{n})$ in $\Z[x,y]$,
and conclude that if $a\in R$ is such that $P(a)=0$ then $(x-a)|P$
in $R[x]$.
\item [{(b)}] Suppose now that $R$ is an integral domain and that $\left\{ a_{i}\right\} _{i=1}^{k}\subset R$
are \emph{distinct} zeroes of $P$. Show that $\prod_{i=1}^{k}(x-a_{i})|P$
in $R[x]$. Give a counterexample when $R$ has zero-divisors.\bigskip{}
\end{lyxlist}
\item [{2.}] (The Vandermonde determinant) Let $\mathcal{V}_{n}(x_{1},\ldots,x_{n})\in M_{n}(\Z[x_{1},\ldots,x_{n}])$
be the $n\times n$ \emph{Vandermonde matrix} $\left(\mathcal{V}_{n}\right)_{ij}=x_{i}^{j-1}$.
Let $V_{n}(\underline{x})=\det\left(\mathcal{V}_{n}(\underline{x})\right)\in\Z[x_{1},\ldots,x_{n}]$
(in other words, the entries of $\mathcal{V}_{n}$ come from the ring
of polynomials in $n$ variables, and hence its determinant is also
in this ring).
\begin{lyxlist}{10.}
\item [{({*}a)\bigskip{}
}] Show that there exists $c_{n}\in\Z$ so that $V_{n}(\underline{x})=c_{n}\prod_{i>j}(x_{i}-x_{j})$.\\
\emph{Hint:} You know $n-1$ zeroes of $V_{n}$, thought of as an
element of $\left(\Z[x_{1},\ldots,x_{n-1}]\right)[x_{n}]$.
\item [{(b)}] Setting $x_{n}=0$ show that $c_{n}=c_{n-1}$, hence that
$c_{n}=1$ for all $n$.
\item [{SUPP}] (Lagrange interpolation) Let $F$ be a field. Show that
for any $\left\{ (x_{i},y_{i})\right\} _{i=1}^{n}\subset F^{2}$ with
the $x_{i}$ distinct there is a unique polynomial $p\in F[x]$ of
degree at most $n-1$ such that $p(x_{i})=y_{i}$. \bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Irreducible polynomials and zeroes }\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{3.}] Let $f\in\Z[x]$ be non-zero and let $\frac{a}{b}\in\Q$ be
a zero of $f$ with $(a,b)=1$. Show that constant coefficient of
$f$ is divisible by $a$ and that the leading coefficient is divisible
by $b$. Conclude that if $f$ is monic then any rational zero of
$f$ is in fact an integer.\bigskip{}
\item [{4.}] Decide while the following polynomials are irreducible
\begin{lyxlist}{10.}
\item [{(a)}] $t^{4}+1$ over $\R$.
\item [{(b)}] $t^{4}+1$ over $\Q$.
\item [{(c)}] $t^{3}-7t^{2}+3t+3$ over $\Q$.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{5.}] Show that $t^{4}+15t^{3}+7$ is reducible in $\Z/3\Z$ but
irreducible in $\Z/5\Z$. Conclude that it is irreducible in $\Q[x]$.
\end{lyxlist}
\begin{center}
\textbf{Derivations and differential rings}\bigskip{}
\par\end{center}
\begin{itemize}
\item Differential rings will be a source of some advanced examples in this
course covered only in problem sets. It's ok to skip this material.
\item In a ring $R$ for any $n\in\Z_{\geq0}$ we identify $n=\overbrace{1_{R}+\cdots+1_{R}}^{n}$
(and similarly for $-n$).
\end{itemize}
\begin{lyxlist}{10.}
\item [{6.}] Let $R$ be a ring, and let $S$ be an $R$-algebra (a ring
with a compatible structure as an $R$-module). A \emph{derivation}
on $S$ is an $R$-linear map $\partial\colon S\to S$ such that $\partial(fg)=\partial f\cdot g+f\cdot\partial g$
for all $f,g\in S$. A \emph{differential $R$-algebra} is a pair
$\left(S,\partial\right)$ with $S,\partial$ as above.
\begin{lyxlist}{10.}
\item [{(a)}] Call $f\in S$ \emph{constant} if $\partial f=0$. Show that
the set of constants is a subring containing the image of $R$ in
$S$.
\item [{(b)}] Show that $\partial(f^{n})=nf^{n-1}\partial f$ for positive
$n$, and also for negative $n$ if $f$ is invertible (hint: apply
$\partial$ to $f^{n}\cdot f^{-n}=1$). Conclude that if $S$ is a
field then the set of constants is a subfield, the \emph{field of
constants.}
\item [{(c)}] Show that for any open interval $I\subset\R$, $\frac{d}{dx}$
is a derivation on the $\R$-algebra $C^{\infty}(I)$.
\item [{(d)}] Show that for any ring $R$, $\partial\left(\sum_{n=0}^{\infty}a_{n}x^{n}\right)\eqdef\sum_{n=1}^{\infty}na_{n}x^{n-1}$
defines a derivation on $R[[x]]$, the \emph{formal derivative}. Show
that if $R$ is an integral domain then the constants of $\partial$
are exactly the constant series $a_{0}+\sum_{n=1}^{\infty}0x^{n}$.
\item [{SUPP}] Let $\partial_{1},\partial_{2}$ be derivations on $S$
and let $\alpha\in R$. Show that $f\mapsto\alpha\partial_{1}f$ and
$f\mapsto\partial_{1}\partial_{2}f-\partial_{2}\partial_{1}f$ are
derivations. This makes the set of derivations into a \emph{Lie algebra}
over $R$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{Supplementary Problems I: Review of ideals}
\par\end{center}
Fix a ring $R$.
\begin{lyxlist}{10.}
\item [{A.}] (Working with ideals)
\begin{lyxlist}{10.}
\item [{(a)}] Let $\mathcal{I}$ be a set of ideals in $R$. Show that
$\bigcap\mathcal{I}$ is an ideal.
\item [{(b)}] Given a non-empty $S\subset R$ show that $(S)\eqdef\bigcap\left\{ I\mid S\subset I\ideal R\right\} $
is the smallest ideal of $R$ containing $S$.
\item [{(c)}] Show that $(S)=\left\{ \sum_{i=1}^{n}r_{i}s_{i}\mid n\geq0,\,r_{i}\in R,\,s_{i}\in S\right\} $.
\item [{(d)}] Let $a\in R^{\times}$. Show that $a$ is not contained in
any proper ideal.\\
\emph{Hint}: Show that $a\in I$ implies $1\in I$.\bigskip{}
\end{lyxlist}
\item [{B.}] (Prime and maximal ideals) Call $I\ideal R$ \emph{prime }if
whenever $a,b\in R$ satisfy $ab\in I$, we have $a\in I$ or $b\in I$.
Call $I$ \emph{maximal} if it is not contained in any proper ideal
of $R$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $R$ is an integral domain iff $(0)=\left\{ 0\right\} \ideal R$
is prime.
\item [{(b)}] Show that $I\ideal R$ is prime iff $R/I$ is an integral
domain.
\item [{(c)}] Show that $R$ is a field iff $(0)$ is its unique ideal
(equivalently, a maximal ideal).
\item [{(d)}] Use the correspondence theorem to show that $I$ is maximal
iff $R/I$ is a field.
\item [{(e)}] Show that every maximal ideal is prime.\\
\emph{Hint:} Every field is an integral domain.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\end{document}