1.Why the object requires prime number p? 2.Why the p-adic norm of x is defined by [tex]|x|_{p}=p^{-m}[/tex]([tex]x=\frac{p^{m}r}{s}[/tex]),not [tex]|x|_{p}=p^{m}[/tex]? 3.[tex]Q_{p}\subset R[/tex] or [tex]R \subset Q_{p}[/tex]? 4.What is the difference between p-adic and l-adic? what is the letter "l" stands for?
1) You can still form the p-adic integers when p is not prime. However, [itex]|x|_p[/itex] wouldn't be a norm ([itex]|xy|_p=|x|_p|y|_p[/itex] wouldn't be true), and the p-adic integers will not be an integral domain (there are zero divisors). 2) the alternative you suggest wouldn't satisfy |x+y|<=|x|+|y|, so it isn't a norm. 3) neither. Q < R and Q < Q_p. 4) l-adic is the same as p-adic, if l=p ! l is just some prime number.
Thank you. One more question here is: What is the relationship between p-adic field and Galois field?
They're quite different. A Galois field is just any field with a finite number of elements; the p-adics form an an infinite field for each prime p.
can you define a p-adic integral of any function f(x) where x- is always a p-adic number Can you define a p-adic differentiation ? in similar manner Is there any relationship between the q-analogue of a function and the p-adic set of numbers?
In arithmetic geometry, one usually uses the letter, p, to denote the characteristic of a base field and "l" for a prime number different from the char. For example, l-adic etale cohomology. p-adic crystalline cohomology.
i once heard a guy talking about galois fields. I asked him what the heck it was. he said it is a finite field. So, if a finite field has q elements, then q is a power of some prime p. there is a subfield F_p in it. Z_p=inv.lim. F_p^n.