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Statement of Ostrowski's theorem is incorrect: we have to get an absolute value from the p-adic valuation described -- the valuation itself is not an absolute value.
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):
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,....
<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,.... One can associate with the p-adic valuation an absolute value <math>|n|_p=p^{-v_P(n)}</math>.
By convention, <math>v_p(0)=\infty</math> for all primes p.
By convention, <math>v_p(0)=\infty</math> for all primes p.
* The [[Fundamental Theorem of Arithmetic|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|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 absolute values described above.
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.