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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 8}
\author{William Stein}
\date{Due: Thursday, April 15, 2004}
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\begin{enumerate}
\item Let $k$ be any field. Prove that the only nontrivial valuations
on $k(t)$ which are trivial on $k$ are equivalent to the valuation
(3.3) or (3.4)\edit{fix} of Lecture 16.
\item A field with the topology induced by a valuation is
a topological field, i.e., the operations sum, product, 
and reciprocal are continuous.
\item Give an example of a non-archimedean valuation on a field that
is not discrete.
\item Prove that the field $\Q_p$ of $p$-adic numbers is 
uncountable.
\item Prove that the polynomial $f(x)=x^3 - 3x^2 + 2x + 5$ 
has all its roots in $\Q_5$, and find the $5$-adic valuations
of each of these roots.  (You might need to use
Hensel's lemma, which we haven't discussed in class.
See Appendix C of Cassels, which you may reference.)

\end{enumerate}


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