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\noindent{\bf Math 431 homework}\hfill Due Friday, September 21, 2001

\medskip

\noindent For the remainder of this homework, let $F$ be 
a quadratic field, i.e., a field of degree two over the
rational numbers.  We fix the following notation.
\begin{itemize}
\item $d$ is a square-free, non-square, integer, and
$F = \Q(\sqrt{d})$;
\item $\Z_F = \Z[\delta]$, where $\delta = (1+\sqrt{d})/2$ if $d \equiv 1 \bmod 4$,
and $\delta = \sqrt{d}$ otherwise; 
\item the result of applying the
nontrivial automorphism of $F$ to~$u$ is denoted $u'$; 
\item the norm of $u \in F$ is $N(u) = uu'$, and the trace is $T(u) = u+u'$
\item $D$, the ``discriminant'' of $F$, is $(\delta-\delta')^2$ which is
equal to $d$ or $4d$ according as to whether $d$ is 1 mod 4 or not.
\item $F$ is an ``imaginary'' quadratic field if $d < 0$ and a ``real'' 
quadratic field if $d > 0$.
\end{itemize}

\ex 
An element $u $ of $\Z_F$ is a unit if and only if $N(u) = \pm 1$.
A unit is nontrivial if it is not equal to $\pm 1$.  Find all
nontrivial units in all imaginary quadratic fields.
Find a nontrivial unit in $\Q(\sqrt{3})$.  Find a nontrivial
unit in $\Q(\sqrt{6})$.  Find a nontrivial unit in
$\Q(\sqrt{29})$.

\ex 
Let $\omega = e^{2\pi i/3} = (-1+\sqrt{-3})/2$.  Show that
$\Z[\omega]$ is a Euclidean domain (with respect to the
function $N(a+b\omega) = a^2-ab+b^2$.)  Show 5 is a
prime element in $\Z[\omega]$ but that 3 and 7 are not.
What is the order of the quotient $\Z[\omega]/I$ where
$I$ is the principal ideal $(1-\omega)$.    Same question
for $I = (5)$.

\ex
Let $p$ be an odd prime.  Prove that $(x^p-1)/(x-1) = x^{p-1}+x^{p-2} +
\cdots + x + 1$ is irreducible, so that $[\Q(e^{2\pi i /p}):\Q] = p-1$.
(Hint: consider $f(x+1)$.)  Prove that $[\Q(\cos(2\pi/p):\Q] = (p-1)/2$.
(Hint: Find an automorphism that fixes the cosine.)

\ex
Let $F = \Q(\sqrt{2})$.  Let $u = -1+\sqrt{2}$.  Prove that
$\Z_F^* = \{\pm u^k: k \in \Z\}$.  (By negating and reciprocating
as necessary, we can take $u$ to be a real number bigger than~1,
and by considering $(u-u')^2 = b^2 D$ the number can be bounded
away from~1.)

\ex Find all integers that are simultaneously triangular and square,
such as 1 and 36.  (A little algebraic juggling will lead you to
the equation $x^2-2y^2 = 1$.)

\ex
Suppose that $K$ is an algebraic number field, and that $\alpha$
is integral over~$K$ in the sense that it satisfies a monic
polynomial whose coefficients are in $\Z_K$.  Prove that $\alpha$
is an algebraic integer.

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