A resource for learning p-adic numbers

Question

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

Answer 1

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.

Answer 2

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

Koblitz' book on p-adic analysis.

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.

Answer 3

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

Answer 4

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.

Answer 5

(A somewhat tongue-in-cheek answer.)

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.

Answer 6

Andrew Baker has some free notes up here