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\begin{document}
\centerline{Math 361 problems for Friday, Week 9}
\bigskip

\centerline{Factoring rational integers in a number ring}
\bigskip

For today's class, please work on the following problems.  Some subset of them
will be part of next week's HW.

\begin{prob}
  Let~$f(x)=x^3+bx^{2}+cx+d$ be an irreducible cubic.  Then the \emph{discriminant} of~$f$ is the
  discriminant of $\{1,\theta,\theta^{2}\}$ where $\theta$ is a root of $f$.  It
  turns out to be given by the formula
  \[
    \Delta[1,\theta,\theta^{2}]= b^2c^2-4c^3-4b^3d-27d^2+18bcd.
  \]
  Consider the case $f(x)=x^3+x^2+3$.
  \begin{enumerate}
    \item Prove that~$f$ is irreducible.
    \item Let~$K=\Q(\theta)$ where $\theta$ is a root of $f$.  Compute the discriminant of~$f$, and use it to argue
      that~$\mathfrak{O}_{K}=\Z[\theta]$.  (We proved a theorem a while ago that
      will be useful.)
    \item Factor the ideals~$(p)$ of $\mathfrak{O}_{K}$ into primes for
      $p\in\{2,3,5,7,11,13,17,19\}$.
    \item There is a \href{https://kconrad.math.uconn.edu/blurbs/gradnumthy/disc.pdf}{theorem of Dedekind} that says a rational prime~$p$
      \emph{ramifies} if and only if~$p$ divides the discriminant.  Are your
      results from the previous part of this problem consistent with that
      result?
  \end{enumerate}
\end{prob}

\begin{prob}
  Let $K=\Q(\sqrt{7})$.  For primes $p$ such that~$2\leq p\leq 23$, use the
  theorem introduced in the last class (involving factoring the minimal
  polynomial of $\sqrt{7}$ modulo~$p$) to factor the principal ideal~$(p)$ of
  $\mathfrak{O}_{K}$.  (Take a look at the Wikipedia page on quadratic
  reciprocity for some relevant material.)
\end{prob}
\end{document}
