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\noindent{\bf Math 431 homework}\hfill Due Wednesday, October 10, 2001

\medskip

\ex
Find odd square-free integers $a$ and $b$ such that $(\sqrt{a}+\sqrt{b})/2$
is an algebraic integer.

\ex
Prove that $D(fg) = D(f) D(g) \Res(f,g)^2$ where $D(f)$ denotes
the discriminant of a polynomial~$f$.

\ex
Prove or disprove: If $E$ and $F$ are distinct cubic extensions
of~$\Q$, and $K$ is a number field that contains both, then the
degree of $K$ over~$\Q$ is divisible by~9.

\ex
Prove that the discriminant of the polynomial $f(x) = x^n+ax+b$ is
$$D(f) = (-1)^{n(n-1)/2} n^n b^(n-1) + (-1)^{(n+2)(n+3)/2} (n-1)^{n-1} a^n.$$

\ex
Show that $(1-\zeta_8)^4$ is an associate of 2 in $\Z[\zeta_8]$
where $\zeta_8 = e^{2\pi i/8}$ is a primitive eighth root of unity.

\ex
Show that if $\m_\alpha(x) = x^3-x-2x^2-8$ and $F = \Q(\alpha)$
then 
$$(\alpha^2+\alpha)/2 \in \Z_F \backslash \Z[\alpha].$$

\ex
Let $p$ be an odd prime, $\zeta = e^{2\pi i/p}$, and $F = \Q(\zeta)$.
Find $N_{F/\Q}(\zeta)$.  Find $N_{F/\Q}(\zeta-1)$.

\ex
Let $I$ be a nontrivial ideal in the ring of integers $\Z_F$ of
a number field~$F$.  Prove that there is a nonzero element in
$I \cap \Z$.

\ex
Find an example of nonunique factorization in $\Q(\sqrt{-15}).$
I.e., find an element that factors into irreducibles in two genuinely
different ways.

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