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Homework 6



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

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. 

\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 $-4 \cdot \disc(\ZZ[\sqrt[3]{2}])$?

\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.



2013-05-11 18:33