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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 5: Dirichlet's Units Theorem/Valuations \\
Due in class Monday, October 1
\end{center}

Please submit seven of the following problems.

\head{References} Neukirch, Sections 1.5 and 1.6 (for the units theorem);
Section 2.3 (for valuations).

\head{Jargon watch} A number field is \emph{totally real} if all of its
embeddings are real. Examples: $\QQ$, $\QQ(\sqrt{D})$ for $D>0$,
$\QQ(\cos \frac{2\pi}{n})$.

Given a real quadratic field and a choice of embedding into $\RR$, the
\emph{fundamental unit} is the smallest unit strictly greater than 1;
together with $\pm 1$, the fundamental unit generates the group of units.

\head{Dirichlet's Units Theorem}

\begin{enumerate}
\item
Let $D>1$ be a squarefree integer. Show that the set of solutions of 
Pell's equation $x^2 - D y^2 = 1$ in rational integers $x,y$
is nonempty, and all solutions are of the
form $x + y \sqrt{D} = \pm(u + v \sqrt{D})^n$ for some integer $n$,
where $(u,v)$ is the solution in positive integers minimizing
$u+v\sqrt{D}$. (Hint: relate this to the units theorem in $\QQ(\sqrt{D})$.)
Optional: prove the same statement for $D$ not a perfect square but
no longer squarefree, by reducing to the squarefree case.
\item
Neukirch exercise I.7.2: Check the following table of fundamental units
$\epsilon_1$
for $\QQ(\sqrt{D})$:
\begin{center}
\begin{tabular}{c|cccccc}
$D$ & 2 & 3 & 5 & 6 & 7 & 10 \\
\hline
$\epsilon_1$ & $1+\sqrt{2}$ & $2 + \sqrt{3}$ & $(1 + \sqrt{5})/2$ &
$5 + 2 \sqrt{6}$ & $8 + 3\sqrt{7}$ & $3 + \sqrt{10}$
\end{tabular}
\end{center}
\item
Compute the group of units in the field $\QQ(\sqrt[3]{2})$. (Hint:
we computed on an earlier homework that the ring of integers is
$\ZZ[\sqrt[3]{2}]$.) Optional: use
Magma's built-in function \texttt{UnitGroup} to check your answer.
\item
Neukirch exercise I.7.4: Let $\zeta$ be a primitive $p$-th root of unity, for
$p$ an odd prime. Show that every unit in $\ZZ[\zeta]$ is equal to a
power of $\zeta$ times a unit in $\ZZ[\zeta + \zeta^{-1}]$.
\item 
Neukirch exercise I.7.5: Let $\zeta$ be a primitive $m$-th root of unity, $m \geq 3$.
Show that the numbers $\frac{1-\zeta^k}{1-\zeta}$ for $(k,m) = 1$ are units
in the ring of integers of the field $\QQ(\zeta)$. The subgroup of
the group of units they generate is called the group of
\emph{cyclotomic units}.
\item
Neukirch exercise I.7.6: Let $K$ be a totally real number field (see above),
and let $T$ be a proper nonempty subset of the set of embeddings of
$K$ into $\RR$. Then there exists a unit $\epsilon$ satisfying $0 <
\tau \epsilon < 1$ for $\tau \in T$, and $\tau \epsilon > 1$ for
$\tau \notin T$. (Hint: apply Minkowski's lattice point theorem to the
unit lattice in trace-zero space.)
\end{enumerate}

\head{Valuations}

The following is a fun and surprising application of valuation theory
due to Paul Monsky (\textit{American Mathematical Monthly}, 1970): a square
cannot be divided into an odd number of nonoverlapping triangles of equal area.

\begin{enumerate}
\item
Let $v$ be an exponential valuation on $K$. Prove that $v$ extends to
the extension $L$ of $K$ in the following cases:
\begin{enumerate}
\item[(a)] $L = K[t]/(P(t))$ for an irreducible polynomial $P$;
\item[(b)] $L = K(t)$.
\end{enumerate}
(Hint: for (a), pick $v(t)$ so that the terms of $P(t)$ do not all have
distinct valuation. For (b), attempt to set $v(t) = 1$.)
\item
Suppose a square is divided into nonoverlapping triangles, and that
the vertices of all of the triangles have been colored in red, green
and blue. (Note: a side of the square or
a triangle may contain vertices of other
triangles.) Suppose that none of the sides of the square or the triangles
contains vertices of all three colors (including its endpoints), but
an odd number of sides of the square have one vertex colored red
and one vertex colored blue.
Prove that one of the triangles has vertices of three different colors.
(Hint: parity.)
\item
Suppose the square $[0,1] \times [0,1]$ has been divided into
$m$ nonoverlapping triangles of area $1/m$. Let $K$ be the field
generated over $\QQ$ by the coordinates of the vertices of the triangles.
Prove that there exists an extension $v$
to $K$ of the 2-adic exponential valuation of $\QQ$.
\item
With notation as in the previous problem, color the vertex $(x,y)$ red
if $v(x)>0$ and $v(y)>0$, blue if $v(x) \leq 0$ and $v(x) \leq v(y)$,
and green otherwise. Prove that vertices of three different colors 
cannot be collinear. (Hint: if you're having trouble dealing with $v$,
first try this assuming $K = \QQ$.
\item
With notation as in the previous problem, prove that if a triangle
has vertices of all three colors, its area has negative valuation.
Conclude that $m$ must be even. (Hint: same hint as the previous exercise.)
\end{enumerate}


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