A resource for learning p-adic numbers

I'm looking for a good resource for learning p-adic numbers. I'm familiar with analysis, topology and overall with noncommutative algebra.

• I guess suggestions may vary depending on your background... – Sasha Oct 18 '11 at 5:19
• On a lighter note: @anon posted Drew Aderburg's A Mathematical Seduction on chat a few days ago. – t.b. Oct 18 '11 at 8:30
• @t.b. Hmm, for some reason your comment didn't ping me, and does not show up in my inbox. $*$shrug$*$ – anon Oct 18 '11 at 9:02
• @anon: pings only work for people that already participated in a thread (OP, editors and commenters). See Hendrik's answer here and the links therein for further explanations. – t.b. Oct 18 '11 at 9:12
• Well, Serre's Course in Arithmetic treats them in chapter 2. – isomorphismes May 23 '15 at 18:22

Fernando Gouvêa's $p$-adic Numbers: An Introduction is gentle and enticing. One of the most endearing books I've read.

• I just found a copy of this. I like the sentiment already: "Our aim is sightseeing, rather than a scientific expedition, we will not worry too much if we fail to emphasize a sutle point here and there, nor if our theorems are less general than they could be, nor in fact, if we do not learn all there is to know." Thanks for the reference. – JavaMan Oct 18 '11 at 19:17

The right book, of course, depends on your background.

"P-adic Analysis compared with Real," by Svetlana Katok is a very gentle introduction to p-adic numbers. This text is suitable for an undergrad who has taken some analysis and topology.

"A Course in p-adic analysis," by Alain Robert is a more terse and advanced book on the subject. It is nicely written as well, but it takes much more background to learn from the book.

• I have read Katok's book a couple times and it is very easy and clear. A good book for a beginner. Gouvea's, I have not read, but I got the impression that it would also be good for a beginner. After one of these books, then you can go on to a more difficult book, but you will bring with you a lot more knowledge and intuition. Both of these books are not too long and not too expensive. – Graphth Jan 20 '12 at 13:50

classics available in English, in approximate order of prerequisites required:

Borevich and Shafarevich introduction to theory of numbers.

Cassels' textbook on local fields in the blue LMS series.

Serre's book on local fields.

There are newer books interpolating between these in difficulty, but those were at least until recently the canonized choices.

For more advanced material, ability to read French is indispensable. Works in French on p-adic number theory, analysis, cohomology, differential equations, etc may actually outnumber those in English.

• Hey I hope I can ask you a question. Does Serre's book really treat p-adic numbers, as in can you learn what they are from that book, starting from zero knowledge about them? I just quickly glanced and the search term "p-adic" gives only a few hits.. – Joachim Dec 24 '14 at 16:23
• As implied in the first sentence of the answer, Serre's book is more compressed than the others, and it very quickly reaches advanced material. – zyx Dec 24 '14 at 16:26
• Thanks for the quick answer! – Joachim Dec 24 '14 at 16:27

There is a book called $p$-adic Numbers by Fernando Q. Gouvea which seems nice, as Mariano pointed out while I was typing. I'm not entirely sure that Mariano and Gouvea are different people, I have never seen them in a room together.

A rather different direction is from the tradition of quadratic forms. The setting for the Hasse-Minkowski principle is the $p$-adic numbers, then certain global relations, and so on. So, there are The Arithmetic Theory of Quadratic Forms by B. W. Jones, Integral Quadratic Forms by G. L. Watson, Rational Quadratic Forms by Cassels, The Sensual Quadratic Form by J. H. Conway.

Andrew Baker has some free notes up here

Since you never bothered to specify your personal background, I'll take the liberty that you are familiar with category theory. In which case, one can construct the $p$-adic numbers by taking an appropriate inverse limit. This MathSE post by Arturo would, for (the hypothetical) you (who is familiar with category theory) then, give a good introduction to what $p$-adic numbers are. See also the book reference Arturo gave there.
• What I like about Gouvêa's book is not so much that it tells you what $p$-adics are but why you want them :) – Mariano Suárez-Álvarez Oct 18 '11 at 22:57