I Why is the p-adic order of zero considered infinite?

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The p-adic order of a positive integer n is defined as the highest power of a prime p that divides n. The p-adic order of zero is considered infinite because dividing zero by any prime p does not yield a non-zero remainder, leading to the conclusion that it can be divided an infinite number of times. This concept is linked to the p-adic absolute value, where |0|_p is defined as 0, reinforcing the idea that the p-adic order of zero is infinite. The discussion clarifies that while p-adic order typically relates to prime factorization, defining ord(0) as infinity is logical within the context of p-adic analysis. Overall, the infinite p-adic order of zero emphasizes its unique properties in number theory.
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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
 
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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.
 
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?
 
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?
 
Thank you very much, you put a smile in my face with this very clear sentence. Thanks!
 
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