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\begin{document}

\begin{center}
\textbf{Math 204A (Number Theory), UCSD, fall 2020 \\
Problem Set 7 -- due Thursday, December 3, 2020}
\end{center}

\begin{enumerate}

\item
\begin{enumerate}
\item[(a)]
Let $p$ be a prime number. Prove that for any commutative ring $R$, any ideal $I$, and any $x,y \in R$ such that $x \equiv y \pmod{I}$,
we have $x^p \equiv y^p \pmod{I^p + pI}$.
\item[(b)]
Let $L/K$ be a Galois extension of number fields, let $\frakq$ be a nonzero prime ideal of $\calO_L$, and choose $\pi \in \frakq \setminus \frakq^2$.
Suppose that $\sigma \in \Gal(L/K)_{\mathfrak{q},0}$ satisfies $\sigma(\pi)/\pi \equiv 1 \pmod{\frakq^n}$ for some positive integer $n$.
Prove that $\sigma(\alpha) \equiv \alpha \pmod{\frakq^{n+1}}$ for all $\alpha \in \calO_L$. (Hint: every element of $\calO_L$
can be written as $\beta^p + \pi \gamma$ for some $\beta, \gamma \in \calO_L$, and similarly even if you replace the exponent $p$ with a higher power of $p$.)
\end{enumerate}

\item
Let $K$ be the number field $\QQ(2^{1/4})$. Let $L$ be the Galois closure of $K$ and put $G = \Gal(L/K)$.
\begin{enumerate}
\item[(a)]
Compute the ring of integers $\calO_L$ (e.g., using SageMath).
\item[(b)]
Check that there is a single prime $\frakq$ of $L$ above $2$ and that $\frakq$ is totally ramified over $2$.
\item[(c)]
Compute the groups $G_{\frakq,s}$ for all $s$.
\item[(d)]
Use the answer to (c) to compute the different of $L/\QQ$.

\end{enumerate}

\item
Let $L/K$ be a Galois extension of number fields. Let $\frakp$ be a prime of $K$ lying above the prime $p$ of $\QQ$.
Let $\frakq$ be a prime above $L$. Suppose that $e = e(\frakq/\frakp)$ is not divisible by $p$ (that is, $\frakq$ is \emph{tamely ramified} over $\frakp$).
\begin{enumerate}
\item[(a)]
Show that $G_{\frakq, 1}$ is the trivial group. (Hint: use what we know about the quotients $G_{\frakq,s}/G_{\frakq,s+1}$.)
\item[(b)]
Show that 
\[
\mathcal{D}_{\calO_{L,\frakq}/\calO_{K,\frakp}} = \frakq^{e-1}.
\]
(Hint: first reduce to the case $f(\frakq/\frakp) = 1$.)
\end{enumerate}

\item
Prove that an element $x$ of $\ZZ_2 \setminus 2 \ZZ_2$ is a perfect square if and only if $x \equiv 1 \pmod{8}$.

\item
\begin{enumerate}
\item[(a)]
Prove that $\ZZ\llbracket x \rrbracket/(x-p) \cong \ZZ_p$. This is done in Neukirch, but try it yourself first.
\item[(b)]
Prove that $\ZZ((x))/(x-p) \cong \QQ_p$.
\end{enumerate}

\item
Prove the following facts that were stated without proof in lecture.
\begin{enumerate}
\item[(a)]
If $p$ is a prime and $m$ is a positive integer, then
\[
\varprojlim_n \ZZ/(p^m)^n \ZZ \cong \ZZ_p.
\]
\item[(b)]
If $m_1, m_2$ are coprime integers greater than 1, then
\[
\varprojlim_{n} \ZZ/(m_1 m_2)^n \ZZ \cong \varprojlim_n \ZZ/m_1^n \ZZ \times \varprojlim_n \ZZ/m_2^n \ZZ.
\]
\end{enumerate}

\item
\begin{enumerate}
\item[(a)]
Prove that the field $\RR$ has no automorphisms other than the identity, using the fact that the squares in $\RR$ are precisely the nonnegative elements.
\item[(b)]
Let $p > 2$ be a prime. Show that every element of $\ZZ_p$ congruent to 1 modulo $p^2$ has a $p$-th root, using the binomial series.
\item[(c)]
Optional: For $p > 2$, prove that the field $\QQ_p$ has no automorphisms other than the identity. (See Zulip for hints.)
\end{enumerate}

\end{enumerate}

\end{document}
