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

$\nu_p(m\cdot n)= \nu_p(m) %2B \nu_p(n)~.$
$\nu_p(m%2Bn)\geq \min\{ \nu_p(m), \nu_p(n)\}.$ Moreover, if $\nu_p(m)\ne \nu_p(n)$ , then $\nu_p(m%2Bn)= \min\{ \nu_p(m), \nu_p(n)\}.$

References

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

p-adic order

Properties

See also

References