What are teh p-adic and adeles ?

[zetafunction]

what are teh p-adic and adeles ??

in many pages realted to mathematical physics they use the p-adic and adeles but what they really are ?, it is like taking a real number arithmetic or similar ?

 

[olliemath]

I thought p-adics only had uses in number theory, shows how ignorant I am! They're basically just a different way of extending the integers (or rationals), by adding extra `numbers'..

The p-adic integers are an extension of the usual set of integers. If you've done any modular arithmetic you'll know about the rings $$Z/pZ$$ or (what I'll call) $$Z_p$$ (where p is a prime) (note some people use this symbol for the p-adic integers themselves).
The rings $$Z_{p^n}$$ have the natural structure of a projective system.. that is there are projections $$\phi_n:Z_{p^n}\to Z_{p^{n-1}}$$ with kernel $$\{0,p^{n-1}\}$$ and these projections fit together in a natural way - so we get projections $$\phi_{m,n}=\phi_m\circ\ldots\circ\phi_{n-1}$$ for m>n which satisfy $$\phi_{l,m}\circ\phi_{m,n}=\phi_{l,n}$$ etc. Thus we have a big sequence of rings with arrows from the bigger ones to the smaller ones.. it seems natural to ask if there's a biggest object from which all these projections come.. this is the idea of a projective limit or inverse limit (sorry if you already knew this!)

Each sequence $$(a_n)$$, where each $$a_n$$ belongs to $$Z_{p^n}$$ and $$\phi_n(a_n)=a_{n-1}$$, can be thought of as stemming from an element in the projective limit.. so we define the projective limit to be the ring consisting of these elements with pointwise multiplication and addition. This ring is known as the p-adic integers.

We can think of the integers as sitting inside the p-adics, because they have a coresponding element in each $$Z_{p^n}$$ and the resulting sequence will be in the projective limit.
For instance 12 has the corresponding 3-adic sequence 0,3,12,12,12,.. All positive integers will eventually have a constant sequence, and negative integers do something similar.. for instance -1 has the 2-adic sequence 1,3,7,15,31,.. but there are uncountably many sequences which are not of this form so we've succeeded in adding more numbers (just like reals have decimal expansions these have p-adic expansions).

The p-adic integers have no zero-divisors so we can take the quotient field, which we call the p-adic numbers.