So I was reading the proofs of each axiom from here and I have a couple of places where I am a bit iffy.
- For the first axiom, it seems as though the $p$-adic order is being used as, for want of a better word, a logarithm would be. Why is that in this case? Why would "applying the $p$-adic order" to both sides essentially get rid of the $p$ at the end? Also, what's the purpose of the arguments of $p^{\nu_p^{\Bbb Z}(n)+1}$ not dividing $n$? I don't quite see the relevance of why that might need to be included for the conclusion of axiom 1.
- For axiom 3, they are saying that because you have this extra $p^{\beta}$ term, then that that makes the right hand side bigger? If so, again...why? If not, then what's going on in that axiom?
Sorry for the cluelessness, thanks in advance for any clarification.
Re question 1: The $p$-adic order of an integer $x$ is defined as the highest power of $p$ which divides $x$. Thus $$\nu_p(x) =k$$
is exactly the same as saying $$p^k \mid x {\text{ and }} p^{k+1} \nmid x,$$
and that is what they are checking to compute $\nu_p(mn)$. (Just $p^k \mid x$ alone would only imply $\nu_p(x) \ge k$.)
Re question 2: Maybe it helps to rephrase both the axiom and the proof like this: What they are checking is that if $m,n$ are both divisible by $p^\beta$, then so is $m+n$.