P-adic Order of Zero: Exploring the Assumption

[DaTario]

TL;DR Summary

Hi all, I would like to know why the p-adic order of zero, i.e., the exponent of the highest power of p (prime) that divides 0, is infinite.

best wishes

Hi All,
The p-adic order of a positive integer n is the exponent of the highest power of the prime p that divides n. I would like to know why it is commonly assumed that the p-adic order of zero is infinite.
best wishes,
DaTario

 

[fresh_42]

The p-adic order and the p-adic absolute value are related by $|x|_p=x^{-\operatorname{ord}(x)}$. Of course we want $|0|_p=0$. The absolute value is the more important quantity.

 

[DaTario]

Thank you, fresh 42. So it means that there is no connection with the purely "prime factorization of integers" meaning of the p-adic order. Is it correct?

 

[fresh_42]

Not really. Only the powers of a fixed prime are considered. However, it makes sense to define $\operatorname{ord}(0)=\infty$ anyway: how often can we divide $0$ by $p$ until we get a remainder?

 

[DaTario]

Thank you very much, you put a smile in my face with this very clear sentence. Thanks!