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. 

This number theory-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Number theory Number theory stubs Wikimedia Foundation. 2010. Игры ⚽ Нужна курсовая? p-adic exponential function P-adic quantum mechanics Look at other dictionaries: 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 ord p ( n ) or ν p ( n ). The most important application of… …   Wikipedia p-adic number — In mathematics, and chiefly number theory, the p adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number… …   Wikipedia P-adic quantum mechanics — One may compute the energy levels for a potential well like this one.[note 1] P adic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. It is the application of p adic analysis to quantum… …   Wikipedia p-adic Hodge theory — In mathematics, p adic Hodge theory is a theory that provides a way to classify and study p adic Galois representations of characteristic 0 local fields[1] with residual characteristic p (such as Qp). The theory has its beginnings in Jean Pierre… …   Wikipedia List of first-order theories — In mathematical logic, a first order theory is given by a set of axioms in somelanguage. This entry lists some of the more common examples used in model theory and some of their properties. PreliminariesFor every natural mathematical structure… …   Wikipedia Singly and doubly even — In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. (The former names are traditional ones, derived from the ancient… …   Wikipedia Parity of zero — Zero objects, divided into two equal groups Zero is an even number. In other words, its parity the quality of an integer being even or odd is even. Zero fits the definition of even number : it is an integer multiple of 2, namely 0 × 2. As a… …   Wikipedia List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia Evenness of zero — The number 0 is even. There are several ways to determine whether an integer is even or odd, all of which indicate that 0 is an even number: it is a multiple of 2, it is evenly divisible by 2, it is surrounded on both sides by odd integers, and… …   Wikipedia Multiplicity (mathematics) — In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point. The notion of multiplicity is important to be… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”