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\lhead{\texttt{MATH 204A}}
\chead{}
\rhead{\texttt{Fall 2024}}
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\title{Homework 7}
\author{due \sout{Friday, November 22} Wednesday, November 27, 2024 at 11:59pm}   
\date{}

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The homework is due in Gradescope at the stated time. You may submit handwritten text (scanned from paper or written on a tablet) or a typed PDF (e.g., created using LaTeX). If you choose the handwritten option, you have to also submit a hardcopy in class or put it in my mailbox. \\

\noindent Collaboration and research is fine, as long as you do the following.
\begin{itemize}
\item Try the problems yourself first!
\item Write the solutions in your own words.
\item Acknowledge all sources and collaborators.\\
\end{itemize}

\hrule

\vspace*{0.5cm}




\begin{enumerate}

\item
Let $K$ be the number field $\QQ(2^{1/4})$. Let $L$ be the Galois closure of $K$ and $G = \Gal(L/K)$. 
\begin{enumerate}
\item
Prove that $L$ is the splitting field of the polynomial \[x^8 - 4x^6 + 8x^4 - 4x^2 + 1.\]
\hspace*{-1cm} This is the field \url{https://www.lmfdb.org/NumberField/8.0.16777216.2}.  \\
\hspace*{-1cm} In terms a root $a$ of this polynomial, a $\ZZ-$basis for $\calO_L$ is\\
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$.


\item
Check that there is a single prime $\frakq$ of $L$ above $2$ and that $\frakq$ is totally ramified over $2$.
\item
Compute the groups $G_{\frakq,s}$ for all $s$.
\item
Use the answer to (c) to compute the different of $L/\QQ$.\\

\end{enumerate}



\item
Show that every quadratic extension of $\QQ_2$ is contained in $\QQ_2(\zeta_{24})$. (Hint: first compute $\QQ_2^*/(\QQ_2^*)^2$.) This is a first step towards the Kronecker-Weber theorem, which will be discussed in more detail in Math 204B.\\

\item
Let $L/K$ be an extension of number fields. Let $\frakp$ be a prime ideal of $K$ and suppose that there is a \emph{unique} prime ideal $\frakq$ of $L$ above $\frakp$.
Let $\alpha \in \calO_K$ be an element whose image in $\calO_L/\frakq$ is a primitive element over $\calO_K/\frakp$.
\begin{enumerate}
\item
Let $P(T) \in \calO_K[T]$ be a monic polynomial whose reduction modulo $\frakq$ is the minimal polynomial of $\alpha$ over $\calO_K/\frakp$. Show that there exists $\beta \in \calO_L$
congruent to $\alpha$ modulo $\frakq$ such that $P(\beta) \in \frakq \setminus \frakq^2$. (Hint: use the first-order Taylor expansion of $P$ at $\alpha$, as in the proof of Hensel's lemma.)

\newpage
\vspace*{0.5cm}

\item
For $\beta$ as in (a), prove that $\calO_{L,\frakq} = \calO_{K,\frakp}[\beta]$. (This is Lemma II.10.4 of Neukirch, but try it yourself first.)\\
\end{enumerate}

\item
Let $L/K$ be a Galois extension of number fields of degree $n$. Let $\frakq$ be a prime ideal of $L$ lying above the prime ideal $\frakp$ of $K$.
\begin{enumerate}
\item
Prove that the $\frakq$-adic valuation of the different of $L/K$ is at most $e-1 + v_{\frakq}(e)$ where $e = e(\frakq/\frakp)$. (Hint: reduce to the case where $e = [L:K]$,
then look at the minimal polynomial of an element of $\frakq - \frakq^2$. If you get stuck, see Theorem III.2.6 of Neukirch.)
\item
Use (a) to prove that for any fixed positive integers $n$ and $D$, there are only finitely many number fields of degree $n$ with discriminant $\pm D$.
Recall that this was the missing step in the proof of the Hermite-Minkowski theorem.\\
\end{enumerate}

\item
\begin{enumerate}
\item
Let $K$ be a field complete with respect to a nonarchimedean absolute value. Let $P(T) \in \calO_K[T]$ be a polynomial, and suppose $\alpha \in \calO_K$ satisfies
\[
|P(\alpha)| < |P'(\alpha)|^2.
\]
Define the sequence $\alpha = \alpha_0, \alpha_1, \dots$ by the \emph{Newton-Raphson recursion}
\[
\alpha_{n+1} = \alpha_n - \frac{P(\alpha_n)}{P'(\alpha_n)}.
\]
Prove that this sequence converges to a root of $P$ in $K$. (Hint: show that $|P'(\alpha_{n+1})| = |P'(\alpha_n)|$ but $|P(\alpha_{n+1})|$ is much smaller than $|P(\alpha_n)|$.)
\item
Apply (a) to recover the fact that any element of $\ZZ_2$ congruent to 1 modulo 8 is a perfect square, then formulate an analogous statement
for any finite extension $K$ of $\QQ_2$.\\
\end{enumerate}

\item
\begin{enumerate}
\item
Let $K$ be a field equipped with (but not necessarily complete with respect to)
a nonarchimedean absolute value $|\cdot|_K$. Let $K(T)$ be the fraction field of the polynomial ring $K[T]$. Prove that there exists a unique absolute value $|\cdot|_G$ on $K(T)$ such that for all $f = \sum_n f_n T^n \in K[T]$,
\[
|f|_G = \max\{|f_n|_K: n =0,1,\dots\}.
\]
This is called the \emph{Gauss norm} on $K(T)$.
\item
Let $L$ be any field containing $K$, not necessarily a finite extension. Prove that there exists at least one extension of $|\cdot|_K$ to an absolute value on $L$. (Hint: using Zorn's lemma, reduce to the case where $L$ is generated over $K$
by a single element $\alpha$. If $\alpha$ is algebraic over $K$, use what we proved in class; otherwise, use (a).)\\
\end{enumerate}




\end{enumerate}







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