Changes

* p-adic values satisfy the [[Archimedes|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 (which some mathematicians view as the "prime at infinity") 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.