p-adic order

In number theory, for a given prime number p, the p-adic order or additive p-adic valuation of a number n is the highest exponent ν such that p^ν divides n. It is commonly abbreviated ν_p(n). The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.

Properties

• $\nu_p(m\cdot n)= \nu_p(m) + \nu_p(n)~.$
• $\nu_p(m+n)\geq \min\{ \nu_p(m), \nu_p(n)\}.$ Moreover, if $\nu_p(m)\ne \nu_p(n)$, then ν_p(m + n) = min{ν_p(m),ν_p(n)}.

See also

• Valuation (mathematics)
• Multiplicity (mathematics)
• Ostrowski's theorem

References

• David S. Dummit; Richard M. Foote (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9. 

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