p-adic numbers

[kowalski]

I'm reading the book "Numbers" by Ebbinghaus et al. (Springer Verlag); I can't understand what's the main idea about "p-adic numbers", and what kind of problems can be solved with this sistem of numbers. Can you explain it to me in (as simple as possible...) few words?.

[g_edgar]

Often used in number theory. There is one completion of the rational numbers for each prime p ... plus the usual completion, the real numbers, which is taken to correspond to a fictitious prime called "infinity".

[kowalski]

i. Do you mean that there are infinite "real numbers, say R_p" (one for each prime p) apart from the usual reals R?. Why are they interesting, what propierties does they have?.

ii. Can you give a (no too esoteric) example of the use of p-adics in Number Theory?. Do they solve 'real' problems, or problems originated in their own existence?.

iii. Any applications out of Number Theory? (math or physics?). [As there are very useful applications of complex, quaternionic and octonions in (theoretical) physics].

[g_edgar]

i. Do you mean that there are infinite "real numbers, say R_p" (one for each prime p) apart from the usual reals R?.

Yes, except they are not called "real numbers" they are called "p-adic numbers". Probably it is more common to write \mathbb{Q}_p for the p-adic numbers, and then maybe correspondingly write \mathbb{Q}_\infty for the real numbers.

Why are they interesting, what propierties does they have?.

ii. Can you give a (no too esoteric) example of the use of p-adics in Number Theory?. Do they solve 'real' problems, or problems originated in their own existence?.

Why not do some reading about them?

iii. Any applications out of Number Theory? (math or physics?).

Google Scholar finds about 400 articles in physics published during 2009 with "p-adic" in the title or abstract. (And only 128 articles for "octonions".)

[Petek]

ii. Can you give a (no too esoteric) example of the use of p-adics in Number Theory?. You could start with the Hasse-Minkowsky Theorem (http://en.wikipedia.org/wiki/Hasse-Minkowski_theorem).

[kowalski]

Thank you very much, g_edgar and Petek. I will tray this references. I'm very surprised to know that there are such a number of papers in physics with 'p-adics' in their title !. K.