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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Problem Set 7: Local Fields/Ramification \\
Due \emph{in Frank Calegari's mailbox} Monday, October 15
\end{center}

This week only: please make any extension requests directly to Frank.
His email is \texttt{fcale@math.berkeley.edu}.

Please submit seven of the following problems.

\head{Jargon watch}
Given a field $K$ and a fixed exponential valuation on $K$, the
\emph{value group} is the image of the exponential valuation in $\RR$.

A field with a discrete valuation
is called \emph{henselian} if the statement of Hensel's Lemma
holds over the field, whether or not the field is complete. For the purposes
of this problem set, you may use the fact that a field $K$ is henselian if and
only if the following weak form of Hensel's Lemma is true: if $P(x)$
is a polynomial over the valuation ring $R$ of $K$ and
$\alpha$ is a simple root of $P$ over the residue field of $R$, then
there is a root of $P$ in $R$ that reduces to $\alpha$. See Section~II.6
for more on henselian fields.

Throughout the set, let $\zeta_n$ denote an $n$-th root of unity.

\head{More on local fields}

\begin{enumerate}
\item
Determine all of the quadratic extensions of $\QQ_p$. (Your answer will
have a different form for $p=2$.)
\item
Determine the degree of $\QQ_p(\zeta_n)$ over $\QQ_p$, for $p$ a prime
not dividing $n$.
\item
Let $K$ be the maximal unramified extension of $\QQ_p$. Prove that $K$
is henselian. (Hint: reduce to a finite extension of $\QQ_p$.)
\item
Let $R$ be the ring of power series over $\CC$ that converge in some
neighborhood of 0. Prove that the fraction field of $R$ is henselian.
\item 
Neukirch II.6.1 (modified): Let $f(x) = x^n + \sum_{i=0}^{n-1} f_i x^i$
be a monic polynomial
of degree $n$ over a local field $K$, which factors completely over $K$
with roots $r_1, \dots, r_n$.
Prove that for any $\epsilon>0$, there exists $\delta>0$ such that
if $g(x) = x^n + \sum_{i=0}^{n-1} g_i x^i$ is a monic polynomial of
degree $n$ such that $|f_i-g_i| < \delta$, then $g$ has $n$ roots
$s_1, \dots, s_n$ in $K$, which can be labeled so that
$|r_i-s_i| < \epsilon$ for $i=1, \dots, n$. That is, the roots of $f$
vary continuously with the coefficients.
\end{enumerate}

\head{Ramification}

\begin{enumerate}
\item
Prove that a finite extension of $\QQ_p$
has only one unramified extension of any given degree. (Hint:
recall that a finite field has only one extension of any given degree.)
\item
Show that $\QQ_p(\zeta_{p}) = \QQ_p(\pi)$ for $\pi$ a $(p-1)$-st root
of $-p$.
(Hint: both extensions are tamely ramified.)
\item
Neukirch II.7.1: The maximal unramified extension of $\QQ_p$ is obtained
by adjoining all roots of unity of order prime to $p$.
\item
Neukirch II.7.3: Let $K$ be a local field and
$L/K$ a totally and tamely ramified extension, and let $\Gamma$
and $\Delta$ be the value groups of $K$ and $L$, respectively.
Show that the subfields of $L$ containing $K$ are in one-to-one correspondence
with the subgroups of $\Delta/\Gamma$.
\item 
Let $P(x)$ be a monic irreducible polynomial over $\ZZ_p$. 
Suppose the discriminant of $P(x)$ is not divisible by $p$. Prove that
the field $\QQ_p[x]/(P(x))$ is unramified over $\ZZ_p$.
\item
For $p$ odd, prove that
$\QQ_p(\zeta_n)$ is ramified over $\QQ_p$ if and only if $p$ divides $n$.
Optional: what happens if $p=2$?
(Hint: see the end of Section~II.7.)
\end{enumerate}

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