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\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 8: Ramification and Splitting \\
Due in class Monday, October 29
\end{center}

Please submit six of the following problems.

\head{Extension of Valuations}

\begin{enumerate}
\item
Neukirch exercise II.8.3: How many extensions to $\QQ(\sqrt[n]{2})$ does the
archimedean absolute value $|\cdot|$ of $\QQ$ admit?
\end{enumerate}

\head{Ramification and Splitting}

\begin{enumerate}
\item
Neukirch I.8.3: If a prime ideal $\gothp$ of a number field
$K$ is totally split in the extensions $L/K$ and $L'/K$, then it is also
totally split in the compositum $LL'/K$.
\item
Write a Magma program to determine the splitting behavior (i.e., the number
of prime factors and the corresponding pairs $(e,f)$) of primes of $\QQ$
in the ideal $\QQ(\sqrt[3]{2})$. (Hint: relate the splitting
to the factorization of the polynomial $x^3-2$ modulo $p$.) 
Optional: compute some examples and
look for patterns in the data; it may help to separate primes based on
$p \pmod{3}$.
\end{enumerate}

\head{Interlude on Quadratic Reciprocity}

\begin{enumerate}
\item
For $n = \prod_i p_i^{e_i}$ odd and coprime to $a$,
define the Legendre symbol $\legendre{a}{n}$ as
$\prod_{i} \legendre{a}{p_i}^{e_i}$. Formulate a version of quadratic 
reciprocity (Theorem 8.6) for this generalized Legendre symbol,
and use this to describe a version of the Euclidean algorithm to compute
Legendre symbols \emph{without} factoring into primes.
\item
Neukirch II.8.8: Let $a_n = \frac{\epsilon^n-\epsilon^{\prime n}}{\sqrt{5}}$,
where $\epsilon = \frac{1+\sqrt{5}}{2}$, $\epsilon' = \frac{1-\sqrt{5}}{2}$,
so that $a_n$ is the $n$-th Fibonacci number. If $p$ is a prime number
other than 2 or 5, prove that $a_p \equiv \legendre{p}{5} \pmod{p}$.
\end{enumerate}

\head{The Relative Discriminant}

\begin{enumerate}
\item
Neukirch defines the relative discriminant of an extension $L/K$ of number
fields as the ideal generated by $\Disc(\alpha_1, \dots, \alpha_n)$ for
all choices $\alpha_1, \dots, \alpha_n$ of a basis of $L/K$ consisting
of elements of $\gotho_L$. Show that this coincides with the definition
I gave in class. (Hint: you might want to peek ahead to Section III.2.)
\item
If $L/K$ is an extension of number fields, prove that the ideal
$(\Disc(L)/\Disc(K))$ (the quotient of the absolute discriminants)
is equal to the absolute norm of the
relative discriminant $\Disc(L/K)$. (Hint: use the definition of the relative
discriminant from class.)
\item
Prove that if $P(x)$ is an irreducible monic polynomial over $\QQ$ whose
discriminant $D$ is squarefree, then $\QQ[x]/(P(x))$ is unramified over
$\QQ(\sqrt{D})$.
\item
Compute the relative discriminant $\Disc(\QQ(\zeta_p)/\QQ)$, using 
the fact that the valuation ring of $\QQ_p(\zeta_p)$ equals $\ZZ_p[\zeta_p]$,
proved in class (and in Section II.7 of Neukirch). Use this result to
prove that the ring of integers of $\QQ(\zeta_p)$ equals $\ZZ[\zeta_p]$.
Optional: peek ahead to Section I.10 and compare the proof given there.
\end{enumerate}

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