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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 4: Lattices\\
Due in class Monday, September 24
\end{center}

\head{References} Neukirch, Sections 1.5 and 1.6.

Please submit \emph{nine} problems. Do not include both of the final
two (Magma-based) problems.

\head{Leftover Problem on Minkowski Space}
\begin{enumerate}
\item (Hermite's Theorem) Prove that for any positive integer $N$,
there are only finitely many number fields $K$ (of all degrees) such that
$|\Disc(K)| \leq N$. (Hint: use one of last week's homework problems
to deduce a lower bound on the discriminant.)
\end{enumerate}

\head{Class Numbers}

\begin{enumerate}
\item
Neukirch I.6.3: Show that in every ideal class of a number field $K$
of degree $n$ and signature $(r,s)$,
there exists an integral ideal $\gotha$ of absolute norm
at most $\frac{n!}{n^n} \left( \frac{4}{\pi} \right)^s \sqrt{|\Disc(K)|}$.
(Hint: imitate the proof given in class, but replace the invocation of
Minkowski with a result from last week's homework.)
\item
Use the previous exercise to deduce that every ideal class of the number
field $\QQ[x]/(x^5-x+1)$ contains an ideal of absolute norm at most 3.
(Hint: first compute the discriminant of the polynomial $x^5-x+1$.
You may use Magma for that step if you wish.)
\item
Prove that the number field $\QQ[x]/(x^5-x+1)$ contains no ideals of norm
2 or 3. Conclude from the previous exercise that it has class number 1.
(Hint: prove that if there is an ideal $\gotha$ of prime absolute norm $p$,
then $x^5-x+1$ must have a root modulo $p$.)
\item
Neukirch I.6.6: Let $\gotha$ be an integral ideal of $K$ and $\gotha^m
= (a)$. Show that $\gotha$ becomes a principal ideal in the field
$L = K(\sqrt[m]{a})$, in the sense that $\gotha \gotho_L = (\alpha)$
for some $\alpha$.
\item
Neukirch I.6.7: Show that, for every number field $K$, there exists a finite
extension $L$ of $K$ such that every ideal of $K$ becomes a principal
ideal in $L$.
\end{enumerate}

\head{Class Groups of Quadratic Fields}

The class groups of quadratic fields were first studied by Gauss, before
the language of algebraic number theory had been developed. He studied them
in terms of binary quadratic forms, which we now take a look at.

Throughout this section, $D$ will denote a 
\emph{fundamental discriminant}, i.e., a number of one of the
following forms:
\begin{enumerate}
\item[(a)] $m$, where $m$ is squarefree and congruent to 1 mod 4;
\item[(b)] $4m$, where $m$ is squarefree and congruent to 3 mod 4;
\item[(c)] $8m$, where $m$ is squarefree and odd.
\end{enumerate}

A \emph{binary quadratic form} of discriminant $D$ is an expression of the
form $ax^2 + b x y + c y^2$ where $b^2 - 4ac = D$. Two binary quadratic
forms $ax^2+bxy + cy^2$ and $a'x^2+b'xy+c'y^2$
are said to be \emph{equivalent} if there exists an integer matrix
$\begin{pmatrix} p & q \\ r & s \end{pmatrix}$ of determinant 1 such that
$a(px+qy)^2 + b(px+qy)(rx+sy) + c(rx+sy)^2 = \pm (a'x^2+b'xy+c'y^2)$.

\begin{enumerate}
\item
Put $K = \QQ(\sqrt{D})$.
Show that the map
\[
ax^2 + bxy + cy^2 \mapsto \ZZ a + \ZZ \frac{b + \sqrt{D}}{2}
\]
of binary quadratic forms to ideals of $\gotho_K$ induces a bijection
between equivalence classes of forms and ideal classes of $\gotho_K$.
\item
Use the previous exercise to define a multiplication operation on
equivalence classes of
binary quadratic forms that translates into the multiplication of
ideal classes.
\item
Prove that every binary quadratic form is equivalent to a form
$ax^2 + bxy + cy^2$ where $0 < a \leq |c|$ and $-a/2 < b \leq a/2$.
Such a form is called \emph{reduced}.
Optional: show that we can choose the form so that
there do not
exist $x,y \in \ZZ$ such that $0 < |ax^2+bxy+cy^2| < a$.
\item
Suppose $ax^2+bxy+cy^2$ is a reduced binary quadratic form of discriminant
$D$. Prove that the corresponding ideal class has order 2 in the class group
if and only if $D$ is even and $b=0$, or $D$ is odd and $a=b$. (Hint:
the inverse of the class corresponding to $ax^2+bxy+cy^2$ corresponds to
$ax^2-bxy+cy^2$. You may also use the optional clause of the previous
problem.) 
\item
(A theorem of Gauss)
Prove that the largest power of 2 dividing the class number
of $\QQ(\sqrt{D})$ is $2^{n-1}$,
where $n$ is the number of distinct prime factors of $D$.
\item 
Use Magma to find all the imaginary quadratic fields $\QQ(\sqrt{-D})$
for $D \leq 10^5$ with class number 2. (Hint: you won't have to write your
own function to compute the class number.) In fact, the complete list of
imaginary quadratic fields of a given class number is now provably known
for class numbers up to about 24.
\item
Use Magma to compute the class numbers of the real quadratic fields 
$\QQ(\sqrt{D})$ for $D \leq 10^5$, or more if you can. (Hint: see previous
hint.) Look at how many of the fields with $D \leq N$ have class number 1,
as a function of $N$, and make a conjecture about how this number depends
on $N$ for $N$ sufficiently large. (This conjecture will be unsolved!)
\end{enumerate}


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