Understanding P-adic Numbers: A Primer

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Discussion Overview

The discussion revolves around the concept of p-adic numbers, their properties, applications in number theory, and potential uses in other fields such as physics. Participants seek clarification on the fundamental ideas behind p-adic numbers and their relevance to various mathematical problems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants explain that p-adic numbers are a completion of the rational numbers for each prime p, alongside the real numbers corresponding to a fictitious prime called "infinity".
  • There is a question about whether there are infinite p-adic numbers and what properties make them interesting.
  • Participants inquire about practical examples of p-adic numbers in number theory, specifically asking if they address real-world problems or are more abstract in nature.
  • One participant suggests looking into the Hasse-Minkowsky Theorem as a starting point for understanding p-adic applications in number theory.
  • There is mention of a significant number of physics articles referencing p-adic numbers, indicating potential applications beyond number theory.

Areas of Agreement / Disagreement

Participants express curiosity about p-adic numbers and their applications, but there is no consensus on their properties or the extent of their usefulness in various fields. Multiple viewpoints and questions remain unresolved.

Contextual Notes

Some limitations include the lack of detailed explanations about the properties of p-adic numbers and their applications, as well as the dependence on specific definitions of terms used in the discussion.

Who May Find This Useful

This discussion may be useful for individuals interested in number theory, mathematical concepts related to p-adic numbers, and their applications in physics and other areas of mathematics.

kowalski
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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?.
 
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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".
 
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].
 
kowalski said:
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".)
 
kowalski said:
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.
 
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.
 

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