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Given a prime number p the '''p-adic value''' is the function, denoted <math>v_p</math> which takes as its argument a natural number n and returns the power of p appearing in the prime factorization of that number (equivalently, the highest power of p which divides n). For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,.... <math>v_p(0)</math> is defined by convention to be infinity for all prime numbers p.

Given a prime number p the '''p-adic value''' is the function, denoted <math>v_p</math> which takes as its argument a natural number n and returns the power of p appearing in the prime factorization of that number (equivalently, the highest power of p which divides n):

<math>v_p(x)=\max\{n:p^n\mid x\}</math>. For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,....

Here are some properties of p-adic values:

 

Here are some important properties of p-adic values:

* p-adic values convert multiplication into addition (akin to the logarithm function): <math>v_p(xy) = v_p(x) + v_p(y)</math>.

* p-adic values convert multiplication into addition (akin to the logarithm function): <math>v_p(xy) = v_p(x) + v_p(y)</math>.

* p-adic values satisfy the archimedean inequality: <math>v_p(x+y) \le \min\{v_p(x),v_p(y)\}</math>.

* p-adic values satisfy the [[Archimedes|archimedean]] inequality: <math>v_p(x+y) \le \min\{v_p(x),v_p(y)\}</math>.

* Equality holds in the above so long as <math>v_p(x)\ne v_p(y)</math>.

* Equality holds in the above so long as <math>v_p(x)\ne v_p(y)</math>.

* The fundamental theorem of arithmetic can be restated compactly using p-adic values: For all natural numbers n, <math>n=\prod_pp^{v_p(n)}</math> where p ranges over all primes.

* The fundamental theorem of arithmetic can be restated compactly using p-adic values: For all natural numbers n, <math>n=\prod_pp^{v_p(n)}</math> where p ranges over all primes.

* p-adic values can be extended to the rational numbers by defining <math>v_p(x/y)=v_p(x)-v_p(y)</math> for all integers x,y.

* p-adic values can be extended to the rational numbers by defining <math>v_p(x/y)=v_p(x)-v_p(y)</math> for all integers x,y.

* Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real absolute value and the p-adic values.

* Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real absolute value (which some mathematicians view as the "prime at infinity") and the p-adic values.

p-adic values are used most commonly in [[number theory]] and [[algebra]], especially in the theory of [[commutative]] [[ring]]s.

p-adic values are used most commonly in [[number theory]] and [[algebra]], especially in the theory of [[commutative]] [[ring]]s.