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?
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 Quote by 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?
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.

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

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