[top] [TitleIndex] [WordIndex]


Homework 7



\Large\bf Homework 7 for Math 581F\\
Due FRIDAY November 16, 2007

Each problem has equal weight, and parts of problems are worth the
same amount as each other.   

\item For each of the following three fields, determining if there is
  an order of discriminant $20$ contained in its ring of integers:
  K = \Q(\sqrt{5}), \quad K=\Q(\sqrt[3]{2}), \quad\text{and}\ldots
$K$ any extension of $\Q$ of degree $2005$.  [Hint: for the last one,
apply the exact form of our theorem about finiteness of class groups
to the unit ideal to show that the discriminant of a degree $2005$
field must be large.]

\item Compute the class group of $\QQ(\sqrt{-15})$ following a similar
  approach to the computation of the class group of $\QQ(\sqrt{10})$
  in the book.  (Do not do this by typing 1 or 2 lines into Sage, but
  instead compute the Minkowski bound, etc.)

\item Prove that the quantity $C_{r,s}$ in our theorem about finiteness
of the class group can be taken to be $\left(\frac{4}{\pi}\right)^{s} \frac{n!}{n^n}$, as follows (adapted from \cite[pg.~19]{sd:brief}):
Let $S$ be the set of elements
$(x_1,\ldots, x_{n})\in\R^n$ such that
  |x_1| + \cdots |x_{r}| + 2 \sum_{v=r+1}^{r+s}
                    \sqrt{x_v^2 + x_{v+s}^2} \leq 1.
Prove that $S$ is convex and that $M=n^{-n}$,
  M = \max\{ |x_1\cdots x_r\cdot (x_{r+1}^2 + x_{(r+1)+s}^2)\cdots (x_{r+s}^2 + x_n^2)| : (x_1,\ldots, x_n) \in S\}.
[Hint: For convexity, use the triangle inequality and
that for $0\leq \lambda \leq 1$, we have
\lambda\sqrt{x_1^2 + y_1^2} &+ (1-\lambda)\sqrt{x_2^2+y_2^2}\\
&\geq\sqrt{(\lambda x_1 + (1-\lambda)x_2)^2 + 
(\lambda y_1 + (1-\lambda)y_2)^2}
for $0\leq \lambda \leq 1$.  In polar coordinates this last inequality
  \lambda r_1 + (1-\lambda)r_2 \geq 
   \sqrt{\lambda^2 r_1^2 + 2\lambda(1-\lambda) r_1 r_2 \cos(\theta_1 - \theta_2) + (1-\lambda)^2 r_2^2},
which is trivial.  That $M\leq n^{-n}$ follows from the inequality
between the arithmetic and geometric means.
\item Transforming pairs $x_v, x_{v+s}$ from Cartesian to polar coordinates,
show also that $v=2^{r}(2\pi)^s D_{r,s}(1)$, where
  D_{\ell,m}(t) = \int \cdots \int_{\mathcal{R}_{\ell,m}(t)}
       y_1 \cdots y_m dx_1 \cdots dx_{\ell} dy_1 \cdots dy_m
$\mathcal{R_{\ell,m}}(t)$ is given by $x_{\rho}\geq 0$
($1\leq \rho\leq \ell$), $y_{\rho}\geq 0$
($1\leq \rho\leq m$) and 
  x_1 + \cdots + x_{\ell} + 2(y_1+\cdots +y_m) \leq t.
\item Prove that
  D_{\ell,m}(t) = \int_{0}^t D_{\ell-1,m}(t-x)dx
     =\int_{0}^{t/2} D_{\ell,m-1}(t-2y)y dy
and deduce by induction that 
  D_{\ell,m}(t) = \frac{4^{-m}t^{\ell+2m}}{(\ell+2m)!}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 

2013-05-11 18:33