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\title[Homework for Fall 2014 Class Field Theory]{Homework for Fall 2014 Class Field Theory}\maketitle

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  Last Modified \today\\
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\textbf{Some notes on class + lectures:}
\begin{enumerate}
\item \textbf{Expectations}: Taken from Tony Varilly's syllabus: ``In
  my experience as a student, most people do not follow all the
  details of a Math lecture in real time. During lecture, you should
  expect to witness the big picture of what's going on.
You should pay attention to the lecturer's advice on what is important and what isn't. A lecturer
spends a long time thinking on how to deliver a presentation of an immense amount of material;
they do not expect you to follow every step, but they do expect you to go home and fill in the gaps
in your understanding.''

\item Please do all of the following.
  \begin{enumerate}
  \item After each class, review all of the lecture notes.
  \item Before the next class, briefly review again. (Make sure that
    you come to each lecture knowing at least the definitions and
    statements from the last lecture.)
  \item Sit down at least once on your own to attempt the homework.
  \item Later, meet with other students, and approach me, to make more progress on the homework and material from the lectures.
  \end{enumerate}
\end{enumerate}


\newpage

\begin{enumerate}
\itemsep1em
\item \textbf{Topological groups.}  (Added Aug 27;  last modified Aug 27.) \vspace{4pt}

Let $G$ be a topological group with identity element $e$. Prove the following.
% See G. B. Folland, Real analysis, Pure and Applied Mathematics, 11.1
\vspace{5pt}
  \begin{enumerate}
\itemsep.3em


  \item If $U$ is an open neighborhood of $e$ then there exists a
    symmetric open neighborhood $V$ of $e$ such that 
    $V\cdot V\subset U$ (where $V$ is said to be \defi{symmetric} if $V^{-1} = V$).
  \item If $H$ is a subgroup of $G$ then its closure $\overline{H}$ is again a
    subgroup, and if $H$ is normal then $\overline{H}$ is normal.
  \item An open subgroup of $G$ is closed, and a finite-index closed
    subgroup of $G$ is open.
  \item An open subgroup of a compact group has finite index.
  \item If $K,K'\subset G$ are compact sets then so is $K\cdot K'$.
  \item $G$ is Hausdorff if and only if $\{e\}$ is closed in $G$.  (In
    other words, if $G$ is $T_1$ then it is Hausdorff.)

  \end{enumerate}


\item \textbf{Profinite groups, Galois groups, topology, etc.} (Added Aug 25;  last modified Aug 27.) 
\vspace{5pt}


\begin{enumerate}
\itemsep1em

\item  A topological group is said to be \defi{topologically cyclic} (resp. \defi{topologically finitely generated}) if it contains a dense cyclic subgroup (resp. dense finitely generated abelian group). Show that $\Zhat$ and $\Z_p$ are topologically cyclic. Similarly, show that the image of $k[x]$ in $k\llbracket x \rrbracket$ is dense. Show that $G_{\Q}$ is not topologically finitely generated.
% For this question, you really just have to grapple with the basic definitions.

\item  Show that a profinite group is compact, Hausdorff, and totally disconnected (i.e.~ any subset with at least two elements can be represented as the union of two or more disjoint nonempty open subsets). Show that an infinite profinite group is uncountable.
% For this question, you really just have to grapple with the basic definitions.

\item  Show that $\Zhat \cong \prod_p \Z_p$. 
%CRT

\item  Let $G$ be a profinite group. Show that the image of a continuous homomorphism $G \to \Z$ (where $\Z$ has the discrete topology) is trivial. Show that the image of a continuous homomorphism $G \to \C^*$ (where $\C^*$ has the complex topology) is finite. 
% Continuous image of a compact set is compact. 


\item  The topology is really necessary in the infinite Galois correspondence. Give an example of a subgroup $H \subset G_{\Q}$ such that $\Qbar^{H} = \Q$ but $H \neq G_{\Q}$.
% Starting point: consider a countable collection of disjoint quadratic extensions. Take the (closed) subgroup that fixes this. Artfully construct a group containing this other than $G_{\Q}$ which fixes every generator.


\item  Define $\Z_p\{x\}$ to be the completion of $\Z[x]$ with respect to the ideal $(p)$. Show that this is not equal to $\Z_p\llbracket x \rrbracket$. Characterize the elements of $\Z_p\llbracket x \rrbracket$ which lie in $\Z_p\{x\}$. Show that $\Z_p\{x\}$ is isomorphic to the completion of $\Z[x]$ with respect to the maximal ideal $(p,x)$.

\item Let $k \subset L \subset K$ be algebraic extensions and suppose that $K$ is Galois over $k$. Show that $\Aut(K/L)$ is a closed subgroup of $\Aut(K/k)$. 
% hint: L is the union of finite extensions $k \subset L' \subset L$, so $\Aut(K/L)$ is the intersection of all of the $\Aut(K/L')$, which are closed subgroups.

\end{enumerate}

% \newpage

\item \textbf{Valuations and $\Q_p$.} (Added Sept 4;  last modified Sept 9.) 
  \begin{enumerate}
  \item Show that $z \in \Q_p$ has a periodic $p$-adic expansion if and only if it is rational.
%Hint: prove that period ==> rational first and then the converse should be clear.
  \item Show that $\Q_p$ has no continuous field automorphisms other
    than the identity. Does it have any non-continuous automorphisms?
%Hint: Q is dense.
\item The boundary of $\Z_p$ (i.e.~ $\Z_p^*$) is clopen. (Dwork called $\Z_p^*$ the $p$-adic unit tire.)
  \item Classify all closed subgroups of $\Z_p$. (Modified Sept 9.) 
  % \item Give an example of an infinite closed subgroup of $\Z_p$ which is not open.
\item Show that $\Z_p$ is homeomorphic to the Cantor set; $\Q_p$ is homeomorphic to the Cantor set minus any single point.
\item Let $G$ be a topological group. A \defi{lattice} is a discrete subgroup $\Lambda \subset G$ which is free (as an abelian group, so $\Lambda \cong \oplus \Z$ and which is discrete with respect to the induced topology. Show that $\Q_p$ contains no lattices.
\item Let $(K,v)$ be a valued field with $v$ non-Archimedean,  discrete, and non-trivial. Prove that $v = v_{\mm}$ for some ideal  $\mm$ of $\calO_{v}$.
\item (Added Sept 8.) Show that two non-Archimedean norms $|\cdot|_1$ and $|\cdot|_2$ give the same topology on a field $K$ if and only if $|\cdot|_1 = |\cdot|_2^{\alpha}$ for some $\alpha > 0$.
% See Lang II.1
  \end{enumerate}


\item \textbf{Hensel's lemma.} (Added Sept 15;  last modified Sept 17.) 
  \begin{enumerate}
  % \item Give an example of a prime $p$ and an irreducible polynomial $f \in \Z_p[x]$ which is reducible mod $p$.
  \item Use Newton iteration (i.e.~ the proof of Hensel's lemma) to compute the $3$-adic expansion of $\sqrt{7}$.
  \item Show that $u \in \Z_2^*$ is a square (in $\Q_2$) if and only if $u$ is congruent to $1$ mod 8.
% Use the strong form of Hensel's lemma
  \item Suppose $p \neq 2$. Show that $u \in \Z_p^*$ is a $p$th power (in $\Q_p$) if and only if $u$ is congruent to a $p$th power mod $p^2$.
  \item Use the Banach fixed point theorem to give another proof of Hensel's lemma.
% show that $x \mapsto x - f'(x)/f'(a)$ is a contraction on a suitable ball around $a$.
 \item Show that an infinite algebraic extension of a complete field $K$ is never complete. Give an example of something in the completion of $\Q_p^{cyc}$ which is not in $\Q_p^{cyc}$.
 \item For a prime $p$, describe all quadratic extensions of $\Q_p$.
% If this is too hard, just do it for $\Q_p \subset \Q_p^{cyc}$, where you can argue explicitly.

  
\item Let $K$ be a non-Archimedean local field with finite residue field of characteristic $p$. Use Hensel's lemma to prove that if $n$ is a  nonzero integer then $(K^\times)^n$ is open with finite index in  $K^\times$ when $p\nmid n$.  When $\text{char}(K)=0$ use the $p$-adic  logarithm to prove that the same result is true for \emph{any} nonzero  integer $n$.  Deduce that any subgroup of finite index in $K^\times$  is open when $\text{char}(K)=0$.  
%[Hint: an open subgroup of a compact topological group has finite index.]


\item Let $K = \F_p(\!(t)\!)$ Show that $(K^\times)^p$ is closed but  not open in $K^\times$.  

\item For which $a \in \Z$ is $7X^2 = a$ solvable in $\Z_7$? For which $a \in \Q$ is it solvable in $\Q_7$? (Added Sept 17.) 
\item Find all quadratic extensions of $\Q_2$ (up to isomorphism). 
%hint: there are 7
  \end{enumerate}

  \item \textbf{Miscellaneous} (Added Sept 17)  Let $K$ be complete with respect to a non-Archimedean discrete valuation. Let $L$ be a purely inseparable algebraic extension. Show that the valuation extends uniquely to $L$.
% Just look at the definition of purely inseparable


\item \textbf{Newton Polygons} (Added Sept 17;  last modified Sept 17.) 
  \begin{enumerate}
  \item Let $E_n(x) = 1 + x + \frac{x}{2} + \cdots + \frac{x^n}{n!}$. Show that $f$ is irreducible.
% Use the formula for $v_p(n!)$ and Lucas' Theorem 
% go here if stuck http://mattbakerblog.wordpress.com/2014/05/02/newton-polygons-and-galois-groups/
  \item Find two monic polynomials of degree 3 in $\Q_5[x]$  with the same Newton polygon, but one irreducible and the other not. 

  \item Let $f - 1 + a_1x + \cdots + a_{2n}x^{2n}$ be a polynomial such that if $\alpha$ is a root of $f$ then $p\alpha$ is also a root of $f$, with the same multiplicity. What does this imply about the shape of the Newton polygon of $f$? Draw all possible such polygons for $n$ in the range $n = 1,2,3,4$.

  \item Let $f(t) \in \Q_p[t]$ and assume that $f'(t) \in \Z_p[t]$. Let $m$ be the order of vanishing of $f'(t)$ mod $p$ at $0$. Prove that if $m < p-2$, then $f$ has at most $m+1$ zeroes in $p\Z_p$.
% This is Lemma 5.1 of http://www-math.mit.edu/~poonen/papers/chabauty.pdf
  \end{enumerate}


\item \textbf{Structure of units} (Assigned Sept 22;  last modified Sept 17.) 
  \begin{enumerate}
  \item Compute $\Q_5^*/\Q_5^{*3}$, $\Q_2^*/\Q_2^{*2}$, $\Q_2^*/\Q_2^{*3}$, and $\Q_3(\sqrt{3})^*/\Q_2(\sqrt{3})^{*3}$.
  \item Let $K$ be a finite extension of $\Q_p$. Find a formula for
    the index of $K^{*n}\subset K^*$.

  \item For $p > 2$, the isomorphism $\log \colon \Z_p^{(1)} \to \Z_p$ endows
    $\Z_p^{(1)}$ with the structure of a $\Z_p$ module. Given $1 + x
    \in \Z_p^{(1)}$ and $z \in \Z_p$, what is $(1+x)^z$? (without
    reference to $\log$.)

  \item Let $K$ be a finite extension of $\Q_p$ with valuation ring
    $\calO$. Let $n > e/(p-1)$. What is the torsion subgroup of
    $\calO^{*(1)}$? What is the structure of $\calO^{*(1)}$ as a
    $\Z_p$ module?
%hint: See Neurkirch 5.7. Punch line: use the main theorem on finitely
%generated modules over PID's

  \end{enumerate}


\item \textbf{Miscellaneous.} (Assigned Sept 24;  last modified Sept 24.) 
  \begin{enumerate}
 \item Show that $\Qbar_p$ is not complete.
 \item Show that $\C_p := \widehat{\overline{\Q_p}}$ is not spherically complete. (A field $K$ with an absolute value is called \defi{spherically complete} if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty.)
 \item Let $p$ be a prime, $K$ be a finite extension of $\Q$, and $\pp_1,\ldots,\pp_r$ the primes of $\calO_K$ above $p$. 
   \begin{enumerate}
   \item Show that $K \otimes_{\Q} \Q_p \cong \prod K_{\pp_i}$.
   \item Show that $\calO_K \otimes_{\Z} \Z_p \cong \prod (\calO_K)_{\pp_i}\cong \prod \calO_{K_{\pp_i}}$.
   \end{enumerate}
 \item Let $K$ be a finite extension of $\Q_p$, with valuation ring $\calO$. Analyze the zeroes of $\log(1+x)$ on $\calO$ using Newton polygons; show that there are only finitely many in $\calO$, and try to say roughly how many zeroes there are. Let $L = \cup_{(n,p) = 1}\Q_p(\zeta_n)$; how many zeroes does $\log$ have on $L$? (See Koblitz chapter 4 for a precise statement on how Newton polygons work for power series.)
 \item Argue via Newton Polygons that the series $\prod_{i = 0}^{\infty}(1-xp^i)$ has a unique root of every negative integer valuation.
 \end{enumerate}


\item \textbf{Miscellaneous.} (Assigned Sept 29;  last modified Sept 29.) 
  \begin{enumerate}
  \item Show that the maximal tamely ramified \textbf{abelian} extension of $\Q_p$ is finite over the maximal unramified extension of $\Q_p$.

  \item List, with proof, all degree 4 extensions of $\Q_5$.
  \item Prove that the completion of an algebraically closed field is algebraically closed.
  \item Recall that $\Q_p^{\text{tame}} = \cup_{(n,p)=1}\Q_p(\zeta_n,p^{1/n})$. Give an explicit presentation (i.e.~generators and relations) of Gal$(\Q_p^{\text{tame}}/\Q_p)$.
 \end{enumerate}


\item \textbf{Miscellaneous.} (Assigned Oct 1;  last modified Oct 1.) 
  \begin{enumerate}
  \item Compute the lower ramification filtration for $\Q_p(\zeta_{p^n})$ and for $\Q_p(p^{1/(p^n)})$.
  \item For what primes $p$ does $\Q_p$ have an extension with Galois group $S_4$.
% See Neuk II.10, exercise 1 for the statement of the answer.

 \end{enumerate}
 

\item \textbf{Different.} (Assigned Oct 5;  last modified Oct 5.) \\
    Let $L/K$ be a finite extension. Recall that the dual of a fractional ideal $\calU \subset L$ is $^*\calU := \{x : x \in L \, |\, \text{Tr}(x\, \calU) \subset \calO_K\}$
  \begin{enumerate}
  \item Show that $^*\calU$ is a fractional ideal.
  \item Show that $^*\calU \cong \Hom_{\calO_K}(\calU,\calO_K)$. 
  \item Is $^{**}\calU\cong \calU$?
% See Neuk II.10, exercise 1 for the statement of the answer.

 \end{enumerate}



\item \textbf{Local Artin Map.} (Assigned Oct 8;  last modified Oct 8.) \\
  \begin{enumerate}
  \item Assume that a functorial local Artin map exists. Verify that proof from class that finite index subgroups of $K^{\times}$ are in bijection with finite abelian extensions of $K$.
  \item Compute all degree 3 extensions of $\Q_2$.
  \item Prove that a finite extension $K \subset L$ of local fields is totally ramified if and only if the image of $N_{L/K}$ contains a uniformizer.
% This one is easy. For an unramified extension, the norm map is surjective on units.


 \end{enumerate}




% From Frank
% \item ) Exhibit two different cubic ramified extensions of Q3, referring to the Jones-Roberts
% database if you like. (The work is in proving that the fields are not isomorphic.)
% \item 



% \item []\textbf{Local, but not discretely valued, fields}
% \item Show that $\Qbar_p$ is not complete.
% \item Show that $\C_p := \widehat{\Qbar_p}$ is not spherically complete. (A field $K$ with an absolute value is called \defi{spherically complete} if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty.)



% % % \newpage


% Unit theorem.
% \item What is the index of $K^*$ mod $K^{*p}$? Other index problems?

% \textbf{Recently added problems.}\\


% \item Test.


\end{enumerate}


\end{document}

