P-adic values

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Given a prime number p, the p-adic value is the function, denoted , 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,.... One can associate with the p-adic valuation an absolute value .

By convention, for all primes p.

Here are some important properties of p-adic values:

  • p-adic values convert multiplication into addition (akin to the logarithm function): .
  • p-adic values satisfy the archimedean inequality: .
  • Equality holds in the above so long as .
  • The fundamental theorem of arithmetic can be restated compactly using p-adic values: For all natural numbers n, where p ranges over all primes.
  • p-adic values can be extended to the rational numbers by defining 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 absolute values described above.

p-adic values are used most commonly in number theory and algebra, especially in the theory of commutative rings.

Completing the field of rational numbers with respect to p-adic values yiels the field of p-adic numbers.