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

  • I
  • Thread starter Thread starter DaTario
  • Start date Start date
  • Tags Tags
    Zero
AI Thread Summary
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.
DaTario
Messages
1,096
Reaction score
46
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
 
Last edited:
Mathematics news on Phys.org
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!
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top