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Author: William A. Stein
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# Homework 6

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\Large\bf Homework 6 for Math 581F\\
Due FRIDAY November 9, 2007
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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}

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}

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2013-05-11 18:33