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\noindent{\bf Math 431 homework}\hfill Due Friday, November 16, 2001

\ex
Find integers $x$ and ~$y$ such that
$$2x^2 + 2xy + 3y^2 = 1607.$$
Explain your methods.  Brute force techniques are acceptable if
you replace the RHS by $1607\cdot (10^{50}+1)$.
(Numerical hint: in building the norm table, it turns out that
the prime 1607 first appears in row 363.)

\ex
Show that if $f(x) \in \Z[x]$ is ``Eisenstein'' for a prime~$p$
then $p$ does not divide the index $[\Z_F: \Z[\alpha]]$, where
$\alpha$ is a root of~$f$, and $F = \Q(\alpha)$.

\ex
Let $F = \Q(\sqrt[3]{m})$, where $m$ is not a perfect cube.
Show that if $p $ is a prime, not equal to~3, such that
$p$ divides $m$ but $p^2$ does not divide~$m$, then
$p$ does not divide the index $[\Z_F: \Z[\alpha]]$.
Show that if $3$ does divide this index then $m \equiv 0 \bmod 9$
or $m \equiv \pm 1 \bmod 9$.
Is it possible for $3$ to be unramified in~$F$ for a suitable~$m$?
Find the ring of integers in $\Q(\sqrt[3]{289})$.

\ex
Find a nontrivial unit in, and the class number of, $\Q(\sqrt[3]{22})$.

\ex
Find the ring of integers in $\Q(\sqrt{5}, \sqrt{13})$.

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