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\usepackage[all]{xy}
\DeclareMathOperator{\Hom}{Hom}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\F}{{\mathbb F}}

\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}

\newcommand{\Zm}{\Z/m\Z}
\newcommand{\Zn}{\Z/n\Z}

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\begin{center}
  {\large Mathematics 111}\\Spring 2011\\Homework 1
\end{center}


\begin{enumerate}

\item For a positive integer $m$, let $\Zm$ denote the usual ring of
  integers modulo $m$.  We wish to consider the existence of ring
  homomorphisms $\varphi: \Zm \to \Zn$ and their properties.  Note
  that this will also inform us of properties of $\Hom_\Z(\Zm, \Zn)$.

  \begin{enumerate}



  \item For positive integers $m$, $n$ show that if $\gcd(m,n) = 1$,
    then (viewed as set of group or $\Z$-module homomorphisms),
    $\Hom_\Z(\Zm, \Zn) = \{0$\}.



  \item If $m$, $n$ are positive integers, show that if $n \mid m$
    there is a natural surjective ring homomorphism $\varphi: \Z/m\Z
    \to \Zn$.  Show that $\varphi$ induces a group homomorphism
    $(\Z/m\Z)^\times \to (\Zn)^\times$ which is also surjective.


  \end{enumerate}

\item \textbf{Inverse limits.}  Let $I$ be a partially ordered set
  (see Appendix I if unsure of the definition), and for each $i \in
  I$, let $A_i$ be a group (also works in other categories).  Suppose
  that for each $i,j \in I$, $i \le j$ we are given a homomorphism
  $\varphi_{ji}: A_j \to A_i$ satisfying:
  \begin{compactitem}
  \item $\varphi_{ji} \circ \varphi_{kj} = \varphi_{ki}$ whenever $i
    \le j \le k$, and
  \item $\varphi_{ii}$ is the identity map for all $i \in I$.
  \end{compactitem}
  Let $P = \{ (a_i) \in \prod_{i\in I} A_i \mid a_i =
  \varphi_{ji}(a_j) \textrm{ whenever } i \le j\}$; $P$ is denoted $P
  = \varprojlim A_i$, and is called the \textit{inverse} or
  \textit{projective} limit of the $A_i$.

  \begin{enumerate}
  \item Show that $\varprojlim A_i$ is a group under the operations
    inherited from the direct product.



  \item (Universal mapping property).  For each $k \in I$, let
    $\varphi_k: \varprojlim A_i \to A_k$ be the standard projection.
    Suppose that $G$ is any group such that for each $i \in I$ there
    is a homomorphism $\pi_i: G \to A_i$ so that $\pi_i =
    \varphi_{ji}\circ \pi_j$ whenever $i \le j$.  Show that there is a
    unique homomorphism $\pi: G \to \varprojlim A_i$ such that
    $\varphi_i \circ \pi = \pi_i$ for all $i \in I$.

  \end{enumerate}


\item \textbf{Examples of inverse limits.}

  \begin{enumerate}
  \item Let $p$ be a prime, $I = \Z_+$ with the usual partial
    ordering, and put $A_i = \Z/p^i\Z$.  For $i \le j$, let
    $\varphi_{ji}:A_j \to A_i$ be the ring homomorphism defined in 1b.
    The inverse limit of the $A_i$, $\varprojlim \Z/p^i\Z$, is denoted
    $\Z_p$ and called the ring of $p$-adic integers.
    \begin{enumerate}
    \item Show that every element of $\Z_p$ can be written uniquely as
      a formal sum $b_0 + b_1 p + b_2 p^2 + b_3 p^3 + \cdots$ where
      each $b_i \in \{0, 1, \dots, p-1\}$.  Represent each element in
      $A_i = \Z/p^i\Z$ as a least residue $\bmod p$ and express the
      element in its base $p$ expansion.  Characterize the action of
      the maps $\varphi_{ji}$ on these representations.  Note that
      this implies that $\Z_p$ is an uncountable set.



    \item Show that an element $\alpha = b_0 + b_1 p + b_2 p^2 + b_3
      p^3 + \cdots \in \Z_p$ is a unit (in the ring) if and only if
      $b_0 \ne 0$.

    \end{enumerate}

  \item Let $I = \Z_+$ and partially order $I$ by divisibility: that
    is ``$n \le m$'' iff $n \mid m$. Let $A_n = \Zn$ and for $n \mid
    m$, let $\varphi_{mn}: \Z/m\Z \to \Zn$ be the ring homomorphism in
    1b. The inverse limit is denoted $\widehat \Z = \varprojlim \Zn$.

    Now consider a field $F$ and fix an algebraic closure $\overline
    F$ of $F$. Let $I$ denote the set of finite Galois extensions
    $K/F$ where $F \subseteq K \subseteq \overline F$.  Partially
    order $I$ by inclusion, that is $K_1 \le K_2$ iff $K_1 \subseteq
    K_2$.  For each $K \in I$, let $A_K = Gal(K/F)$.

    \begin{enumerate}
    \item For $K \subseteq L$ (elements of $I$), define the natural
      group homomorphisms \newline $\varphi_{LK}: A_L \to A_K$ and
      justify (via Galois theory) that they are surjective group
      homomorphisms.



    \item Now let $F = \F_p$.  Give a more complete description of the
      elements of $I$ and the corresponding $A_i$, and a refined
      notion of the partial order.  Justify that the Galois group of
      $\overline F/ F$, $Gal(\overline F/F) := \varprojlim Gal(K/F)
      \cong \widehat \Z$.


    \item Use this characterization of $\widehat \Z$ to show that that
      for any prime $q$, $\Z_q \subset \widehat\Z$.

    \end{enumerate}
  \end{enumerate}


\end{enumerate}


\end{document}
