# Homework 6

\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}