Homework 6

hw6.pdf

\documentclass{article}
\include{macros}
\voffset=-0.05\textheight
\textheight=1.1\textheight
\hoffset=-0.05\textwidth
\textwidth=1.1\textwidth

\begin{document}
\begin{center}
\Large\bf Homework 6 for Math 581F\\
Due FRIDAY November 9, 2007
\end{center}

Each problem has equal weight, and parts of problems are worth the
same amount as each other.  This homework assignment is short because
of the midterm this weekend. 

\begin{enumerate}
\item 
\begin{enumerate}
\item Give a simple description of the set 
 $$
 X = \{\disc(R) : \text{ $R$ is an order in $\ZZ[i]$ } \}.
$$
\item Is there an order of $\QQ[\sqrt[3]{2}]$ that
has discriminant $-4 \cdot \disc(\ZZ[\sqrt[3]{2}])$?
\end{enumerate}

\item When I was a graduate student Ken Ribet asked me
to determine whether or not the prime $389$ divides the discriminant
of a certain order $T$ generated by an {\em infinite} list
of explicit but hard-to-compute algebraic integers 
$a_2,  a_3,  \ldots$.  Using ``modular symbols'' I computed 
that the characteristic polynomial of $a_2$ is
$$
f = x^{20} - 3x^{19} - 29x^{18} + 91x^{17} + 338x^{16} - 1130x^{15} -
2023x^{14} + 7432x^{13} + 6558x^{12}
$$
$$ \qquad - 28021x^{11} - 10909x^{10} +
61267x^{9} + 6954x^{8} - 74752x^{7} + 1407x^{6} + 46330x^{5} - 1087x^{4}
$$
$$
- 12558x^{3} - 942x^{2} + 960x + 148.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
From this, we see easily that
$$
\disc(f) = 2^{58} \cdot 5^{3} \cdot 211^{2} \cdot 389 \cdot 65011^{2} \cdot
215517113148241 \cdot 477439237737571441.
$$
Is this enough to conclude that the discriminant of $T$
is divisible by $389$?  (Yes or no?  Why or why not?)

\item What is the volume of the real lattice obtained by embedding the
  field $K(\alpha)$, for $\alpha$ a root of $x^{3} - 4x - 2$ in $\R^3$
  via a choice of the embedding from class (that sends $\alpha$ to
  each of the images of $\alpha$ in $\R$)?  Draw a sketch of a
  fundamental domain for this lattice.


\end{enumerate}


\end{document}