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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 9}
\author{William Stein}
\date{Due: Thursday, April 22, 2004}
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\maketitle

There are SIX problems.

\begin{enumerate}
\item In this problem you will compute an example of weak approximation,  like I did in the example in Section 6 of 
lecture 18.\edit{ref}
Let $K=\Q$, let $\absspc_7$ be the $7$-adic absolute value,
let $\absspc_{11}$ be the $11$-adic absolute value, and 
let $\absspc_{\infty}$ be the usual archimedean absolute value.
Find an element $b\in \Q$ such that $\abs{b-a_i}_i<\frac{1}{10}$,
where $a_7 = 1$, $a_{11} = 2$, and $a_{\infty} = -2004$.

\item 
Prove that $-9$ has a cube root in $\Q_{10}$ using the following strategy (this is a special case of Hensel's Lemma, which you can read about in an appendix to Cassel's article).

\begin{enumerate}
\item Show that there is an element $\alpha\in\Z$ such that $\alpha^3\con 9\pmod{10^3}$. 
\item Suppose $n\geq 3$. 
Use induction to show that if $\alpha_1\in\Z$ and 
$\alpha^3\con 9\pmod{10^n}$,  then there exists $\alpha_2\in\Z$ such 
that $\alpha_2^3\con 9\pmod{10^{n+1}}$.
(Hint: Show that there is an integer~$b$ such that
$(\alpha_1 + b\cdot 10^{n})^3 \con 9\pmod{10^{n+1}}$.)
\item Conclude that $9$ has a cube root in $\Q_{10}$.
\end{enumerate}

\item\label{ques:approxfield}  Prove that every finite extension of $\Q_p$ ``comes from''
an extension of~$\Q$, in the following sense.  Given an irreducible polynomial $f\in\Q_p[x]$ there exists an irreducible polynomial $g\in \Q[x]$ such that the fields 
$$\Q_p[x]/(f)$ and $\Q_p[x]/(g)$ are isomorphic.
[Hint: Choose each coefficient of $g$ to be sufficiently
close to the corresponding coefficient of $f$, then use
Hensel's lemma to show that $g$ has a root in $\Q_p[x]/(f)$.]

\item Find the $3$-adic expansion to precision 4 of each root of the following polynomial over $\Q_3$:
$$
  f = x^3 - 3x^2 + 2x + 3 \in \Q_3[x].
$$
Your solution should conclude with three expressions of the form 
$$a_0 + a_1\cdot 3 + a_2\cdot 3^2 + a_3 \cdot 3^3 + O(3^4).$$

\item
\begin{enumerate}
\item Find the normalized Haar measure of the following subset of 
$\Q_7^+$:
$$
U = B\left(28,\frac{1}{50}\right) = 
\left\lbrace x\in \Q_7 : \abs{x-28} < \frac{1}{50}\right\rbrace.
$$
\item
Find the normalized Haar measure of the subset $\Z_7^*$ of
$\Q_7^*$.
\end{enumerate} 


\item Suppose that $K$ is a finite extension of $\Q_p$ and $L$
is a finite extension of $\Q_q$, with $p\neq q$ and assume
that $K$ and $L$ have the same degree.  Prove that
there is a polynomial $g\in \Q[x]$ such that $\Q_p[x]/(g)\isom K$
and $\Q_q[x]/(g)\isom L$.  [Hint: Combine your solution to \ref{ques:approxfield} with the weak approximation theorem.]
 

\end{enumerate}


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