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\noindent{\bf Math 431 Halloween homework}\hfill Due Friday, November 2, 2001

\ex
Show that if $I$ and $J$ are relatively prime ideals in a number ring
and their product is a $k$-th power, then each is a $k$-th power.
Show that if $k$ is relatively prime to the order of the class
group $\Cl(F)$, and $I^k$ is principal, then $I$ is principal.

\bigskip

\noindent From now on, let $F = \Q(\sqrt{-355})$.


\ex
How many ideals are there in $\Z_F$ of norm 
$75245625 = 3^2\cdot 5^4 \cdot 7^3\cdot 13$?

\ex
Find the class group of~$F$.

\ex
Find all integer solutions to the equation
$$y^2+355 = x^3.$$
You may want to show that (a) in any solution $x$ is odd,
(b) consider the equation in principal ideals in $\Z_F$ that
corresponds to a solution and show that the two ideals on the
LHS are relatively prime,
and (c) show that for any solution $(x,y)$, the ideal $(y+\sqrt{-355})$ is
the cube of a principal ideal.

\ex
Show that there is a prime ideal~$P$ such that
$$P = \spn_{\Z}(5,-3+\delta).$$
(The index can be computed immediately, but you have to check
that the RHS is an ideal.)
Show that
$$N(5x+(-3+\delta)y)/N(P) = 5x^2-5xy+19y^2.$$
Show that a number $n$ is represented by this quadratic form if
and only if there is an ideal $I$ of norm~$n$ that is in the
ideal class $[P']$.  Find a representation of 
$40131 = 3^2\cdot 7^3\cdot  13$ by
this form.

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