*Memorizing* proofs doesnâ€™t really do much for you, at least in the long run; instead, you should try to see what makes them tick. First, what is the structure of the argument? What are the main steps, and what are merely details of carrying out those steps? Many proofs at this stage of your studies have just a single main idea, and everything else is details. Secondly, what kinds of details appear over and over? What basic technical tricks keep reappearing? Those are tools that you want to master for your own use.

I think a basic observation is that one can obtain a topology on a ring $R$ simply from the data of an ideal $I \subset R$; this is the $I$-adic topology, where a neighborhood basis at the identity consists of the powers of the ideals $I^i$. The nice thing about doing so is that, in essence, solving a polynomial equation mod a high power of $I$ essentially amounts to finding an "approximate solution" in $R$ itself: the notion of "approximate" solution is made valid by the existence of a topology.

In the case where $I $ is the ideal $p \mathbb{Z}$ inside the ring $\mathbb{Z}$,
then the point of this discussion is that one has translated (purely arithmetic) questions of the form "When does there exist an integer $x$ such that $P(x) \equiv 0 \mod p^N$?" into more analysis-y questions of the form "When does there exist an integer $x$ such that $P(x) $ is close to zero?" So far, we have just moved words around without content, but we can make a big step forward by noting that analysis actually works pretty well only when one has a completeness property (like the completeness property of the real numbers). Without the completeness property of the real numbers, most of ordinary mathematical analysis would fail: try doing integration over the rationals, for instance.

Anyway, one does not usually have completeness of the topology induced on the ring $R$ given by an ideal $I$, but there is a mathematical procedure that enables you to replace $R$ by something where the completeness process is true: this process is called completion. When applied to the integers with the ideal $p \mathbb{Z}$, one gets the ring of $p$-adic integers, and more generally applied to rings of integers in number fields one can get rings of integers in local fields.

Again, one of the things that is nice about completion is that one can in fact do analysis in a reasonable way. A standard example is given by Hensel's lemma, which allows you to lift approximate solutions to exact solutions (in the completion).
The completions turn out to provide information about the integers themselves (the starting point): for instance, it is a general principle that if something is true over $\mathbb{R}$ and all the $p$-adic fields $\mathbb{Q}_p$, then it should be true over $\mathbb{Q}$. The usual example is the Hasse-Minkowski theorem for quadratic forms, but the analog for other polynomials is false.

As for the infinite Galois group: the answer is that it is an inverse
limit of the Galois groups of finite subextensions, and as such inherits a topology in a similar way as the completion.

## Best Answer

It depends on the age of the book.

"

A Course of Modern Analysis" by E. Whittaker, first published in 1902, covers analytic functions, infinite series, Fourier series, Riemann integration etc. - all topics that today we would just call "analysis".On the other hand "

Introduction to Modern Analysis" by S. Kantorowitz, published in 2003, covers measure theory, functional analysis, Banach algebras, spectral theory etc. i.e. a more abstract and generalised treatment of analysis.