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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):

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,....  One can associate with the p-adic valuation an absolute value <math>|n|_p=p^{-v_P(n)}</math>.

<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>.