\documentclass[12pt]{article}
\newcommand{\Z}{{\bf Z}}
\newcommand{\Q}{{\bf Q}}
\newcommand{\R}{{\bf R}}
\newcommand{\m}{{\rm m}}
\newcommand{\disc}{{\rm disc}}
\newcommand{\trr}{{\rm Tr}}
\newcommand{\Res}{{\rm Res}}
\pagestyle{empty}
\newcounter{exercisenum}
\newcommand{\ex}{\medskip\stepcounter{exercisenum}\noindent{\bf \theexercisenum. }}
\begin{document}
\vspace*{-.5in}
\noindent{\bf Math 431 Quiz}\hfill Due Friday, October 12, 2001

\noindent Do any six of the following problems.

\noindent Eight hour time limit; hints available in L305.

\medskip

\ex
Let $\alpha$ be a root of $f(x) = x^3-4x-1$ and let
$F = \Q(\alpha)$.
Prove that $\Z_F = \Z[\alpha]$.  Find a unit in $\Z_F$.
Show that $2\Z_F$ is not a prime ideal (hint: to find
two algebraic integers whose product is in the ideal, you could
choose one of them to be $\alpha+1$.)
Find $\trr_{F/\Q}(\alpha^k)$ for $k = 0,1,2,3$.

\ex
Let $u_0, u_1, \cdots $ be a sequence of integers (i.e.,
elements of~$\bf Z$) such that
$$u_{n+3} - 4 u_{n+1} - u_{n} = 0$$
for all $n \ge 0$.
Prove that there are numbers $a_i$ so that
$$u_n = \sum_{i=1}^3 a_i \left ( \alpha^{(i)} \right )^n$$
where $\alpha$ is as given in the previous problem.
(Hint: consider the first 3 terms and the *** matrix.)
What can you say about 
$$\lim_{n \to \infty} \left | u_n \right | ^{1/n} \quad ? $$

\ex
Describe all solutions to the diophantine equation
$$x^2 + xy - 8 y^2 = 1.$$

\ex
Show that if a polynomial with rational coefficients
has $r_1$ real roots and $r_2$ pairs of complex conjugate
roots, then the sign of its discriminant is $(-1)^{r_2}$.
(Hint: One easy way to do this is to factor the polynomial
over~$\R$, and then use formulae.)  Show that if $F$ is
a number field with $r_1$ real embeddings and $r_2$
pairs of complex conjugate embeddings then the sign of
its discriminant is $(-1)^{r_2}$.

\ex
Show that if $I$ is a nonzero ideal in the ring of integers $\Z_F$
of a number field then $I \cap \Z$ is a nonzero ideal in~$\Z$.

\ex
For each prime ideal $P$ in $\Z[i]$ (see the class notes for 9/21 
for an explicit list)
find, with explanation, the index $[\Z[i]:P]$ of $P$ in $\Z[i]$.

\ex
Let $\alpha$ and $\beta$ be algebraic integers of
degree~$n$ that satisfy
$$\trr_{\Q(\alpha)/\Q}(\alpha^k)
= \trr_{\Q(\beta)/\Q}(\beta^k)$$
for $ 0 \le k \le n$.  Prove that $\alpha$ and $ \beta$ are conjugates.
\end{document}

