扩展：$p$-adic的有理数值

Prove that $v_p(a+b)=\min\{v_p(a),v_p(b)\}$ if $v_p(a)\neq v_p(b)$.

What can we say about $v_p(a+b)$ if $v_p(a)=v_p(b)$?

Prove that $v_p(ab)=v_p(a)+v_p(b)$ for positive rational $a,b$.

We have only defined $p$-adic valuation for positive rational numbers. What about negative rationals? How would you define $v_p(0)$?

(Keep in mind the fundamental concept that when extending a definition, one tries to preserve as many properties as possible.)