$p$-adic Numbers Textbook Suggestions

Question

I would like a challenging and complete introduction to $p$-adic numbers (as I eventually plan to study $p$-adic geometry). I understand that Gouvêa’s book is popular, however, I’m sure that there are lesser-known texts that are high quality as well.

You can check out this post: A resource for learning p-adic numbers. (And maybe also other questions linked there.) — Martin Sleziak, Jul 19 '20 at 15:23

score 3 · Answer 1 · edited Jul 19 '20 at 16:24

3

Serre, Local Fields (or the French original Corps Locaux which as far as I recall has fewer typos).

Schikhof, Ultrametric Calculus.

Fesenko / Vostokov, Local Fields and their Extensions

Cassels, Local Fields.

Some texts on local class field theory (Iwasawa; Neukirch) also contain short, succinct introductions.

edited Jul 19 '20 at 16:24

J. W. Tanner

60,406

answered Jul 19 '20 at 15:36

Torsten Schoeneberg

26,051

What are the prerequisites for Serre’s book? – Adam French Jul 19 '20 at 15:43
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To read Serre you should be comfortable with commutative algebra. If you have never seen $p$-adic numbers before then read Gouvea, Koblitz, Cassels, or Schikhof first. (Note a new edition of Gouvea's book just came out.) Another introductory book is Alain Robert's "A Course in $p$-adic Analysis". After understanding how $\mathbf Q_p$ and their finite extensions behave from one of those books, read Serre or Fesenko/Vostokov. – KCd Jul 19 '20 at 15:54
Contentwise, I guess algebra (i.e. basic Galois theory, group theory, field theory, commutative algebra). More importantly, a determined mind and willingness to fill in many details. Serre is succinct and short because he often leaves out steps which are clear to the experts but not yet to us. Think of that as a good challenge. If you want to study $p$-adic geometry, or anything $p$-adic, at some level people will kind of expect you know Serre's classic. – Torsten Schoeneberg Jul 19 '20 at 15:57

ShBh · Answer 2 · 2020-07-19T16:36:14.863

3

There are some good books I know. I am listing them.

(1) p-adic Numbers: An Introduction by Fernando Q. Gouvêa

(2) A Course in p-adic Analysis by Alain M. Robert

(3) Introduction To p-adic Analytic Number Theory by M Ram Murty

Also, there is a recent open lecture-note Introduction To p-adic Numbers and p-adic Analysis by Andrew Baker in AMS open math notes.

Hope it will help.

edited Jul 19 '20 at 16:36

answered Jul 19 '20 at 16:17

ShBh

6,054

$p$-adic Numbers Textbook Suggestions

2 Answers2