Question about p-adic number

[navigator]

1.Why the object requires prime number p?
2.Why the p-adic norm of x is defined by |x|_{p}=p^{-m}(x=\frac{p^{m}r}{s}),not |x|_{p}=p^{m}?
3.Q_{p}\subset R or R \subset Q_{p}?
4.What is the difference between p-adic and l-adic? what is the letter "l" stands for?

[morphism]

All of your questions are answered on Wikipedia's page on p-adic numbers (http://en.wikipedia.org/wiki/P-adic_number).

[gel]

1.Why the object requires prime number p?
2.Why the p-adic norm of x is defined by |x|_{p}=p^{-m}(x=\frac{p^{m}r}{s}),not |x|_{p}=p^{m}?
3.Q_{p}\subset R or R \subset Q_{p}?
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, |x|_p wouldn't be a norm (|xy|_p=|x|_p|y|_p 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.

[navigator]

Thank you.
One more question here is: What is the relationship between p-adic field and Galois field?

[CRGreathouse]

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.

[mhill]

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?