I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}\_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\mathbb{Q}\_p$ of the rationals with respect to a $p$-adic absolute value. The resulting field is then complete, but has no good reason to be algebraically closed. You can take its algebraic closure, but that is not complete, so then you take the completion of that, and get a field which is both complete, and algebraically closed, denoted by $\mathbb{C}\_p$.


I understand that it is a reasonable desire to have a field extension of $\mathbb{Q}\_p$ that is both complete and algebraically closed; my trouble, however, is getting some sort of grasp on how to picture this object, and to develop any intuition about how it is used. Here are my questions; I'd imagine the answers are related:


1. Am I even supposed to be able to picture it?
2. Is there some way I ought to think of a typical element?
3. Is it worth it, in terms of these goals, to look at the proofs of the assertions in my first paragraph?
4. How is $\mathbb{C}\_p$ typically used? (this question may be too vague, feel free to ignore it!)


Please feel free to answer any or all of these questions.



How to picture $\mathbb{C}_p$?
1. You do whatever works for you. Some people think more algebraically, others more geometrically. I certainly don't know what "to picture" means in this context, but then, I am a more algebraic person, so maybe others will be able to say more. Can you picture $\mathbb{Q}^{ab}$, say? I can't.
2. A typical element is, by definition, represented by a Cauchy sequence of elements of $\overline{\mathbb{Q}}\_p$. Each of the elements in the Cauchy sequence lives in a finite extension of $\mathbb{Q}\_p$, so you can view it in the usual way, as a power series in a uniformiser of that finite extension with coefficients in a finite field. But the field $\overline{\mathbb{Q}}\_p$ itself is not discretely valued, so you cannot pick a common uniformiser for all the numbers in your Cauchy sequence.
3. Yes! In my opinion, that's the only way to get a feel for all the fields involved.
4. That one really is too broad. As you may guess, these fields always come in when you need something $p$-adic that is complete and algebraically closed at the same time. Sometimes, you only need something that is complete and has an algebraically closed residue field. Then, people work with the completion of $\mathbb{Q}\_p^{nr}$. For example, these fields are used all the time in $p$-adic Hodge theory (you will find many introductions if you google) and, consequently, in the theory of Galois representations. To expand on that would require a whole essay, which I'm afraid I am not qualified to write.


