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  \textup{2010} Mathematics Subject Classification} 

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Assignment 1.  Due September 29 in class.  Feel free to assume the results from any earlier questions in doing the later questions.
\begin{enumerate}
\item Show that the inverse limit of an inverse system of nonempty groups is nonempty and forms a group.
\item Show that an infinite profinite group is uncountable.
\item Recall that a topological space $X$ is totally disconnected if the connected components of $X$ are the one-element subsets of $X$.  Show that an inverse limit of totally disconnected topological groups is totally disconnected.  Conclude that $\mathbb{Z}_p$ is totally disconnected.
\item (For the next few questions use the analogy with vector spaces having a basis and try to adapt the proofs here to this setting---it might be a worthwhile exercise to study the corresponding arguments for vector spaces and see if you can adapt them here.)  Recall that if a field $K$ is an extension of a field $k$ then we say that a subset $S$ is algebraically independent if whenever $m\ge 0$ and $s_1,\ldots ,s_m$ are distinct elements of $S$ then if $p(s_1,\ldots ,s_m)=0$ for some polynomial $p(x_1,\ldots ,x_m)\in k[x_1,\ldots ,x_m]$ then $p(x_1,\ldots ,x_m)=0$.  A subset is called a transcendence base if it is a maximally algebraically independent subset.
Let $K$ be a field extension of the field $k$.  Show that $K$ has a transcendence base over $k$.  
\item Show that any two transcendence bases for $K/k$ have the same cardinality.  This cardinality is called the \emph{transcendence degree}.  (Hint: look at the proof that bases for a vector space have the same cardinality and add a few new ideas.)
\item Let $K$ be an extension field of the field $k$ and suppose that ${\rm dim}_k K$ is uncountable and $k$ is countable.  Show that $K$ has uncountable transcendence degree over $k$.  
\item Show that ${\rm dim}_{\mathbb{Q}} \mathbb{Q}_p$ is uncountable and conclude that $\mathbb{Q}_p$ has uncountable transcendence degree over $\mathbb{Q}$.  
\item Urysohn's metrization theorem gives that a second-countable compact Hausdorff space is metrizable.  (Recall that a topological space is second-countable if there exists a countable collection of open subsets such that every open set can be written as a union of elements from this countable collection.)  Suppose that $(S,\le)$ is a countable directed poset and $\widehat{G}$ is the inverse limit of an inverse system $\{G_s\}_{s\in S}$ of finite groups (endowed with the discrete topology).  Show that $\widehat{G}$ (the inverse limit) is metrizable.   (See more on this in questions 11 and 12.)

\item Show that $\mathbb{Z}_p$ has a unique maximal ideal (recall that a ring having this property is called a local ring) given by $p\mathbb{Z}_p$.  (Remember the old trick (well, not much of a trick) to show a maximal ideal is unique: show everything outside the set is a unit.)
\item Show that every nonzero element of $\mathbb{Q}_p$ can be written uniquely as $p^m y$ where $m\in \mathbb{Z}$ and $y$ is in $\mathbb{Z}_p^*$.  
\item Show that we can put a metric $|\, \cdot\, |_p$ on $\mathbb{Q}_p$ in which we define 
$|x|_p =p^{-m}$ if $x$ is nonzero, where we write $x=p^m y$ with $m\in \mathbb{Z}$ and $y$ a $p$-adic unit (that is a unit in $\mathbb{Z}_p$).  (We take $|0|_p=0$.)
\item Show that the restriction of this metric to $\mathbb{Z}_p$ induces the same topology on $\mathbb{Z}_p$ as the profinite topology.  (This relates to question 8---in general, if one has a countable inverse system of finite groups endowed with the discrete topology, one has that it is metrizable and one  can define a distance function in a similar way (it is not unique, of course).  
\end{enumerate}
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