''p''-adic order
In number theory, for a given prime number p, the p-adic order or p-adic valuation of a non-zero integer n is the highest exponent ν such that pν divides n. The p-adic valuation of 0 is defined to be infinity. It is commonly denoted νp(n). If n/d is a rational number in lowest terms, so that n and d are relatively prime, then νp(n/d) is equal to νp(n) if p divides n, or -νp(d) if p divides d, or to 0 if it divides neither. The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.[1]
Definition and properties
Integers
Let p be a prime in ℤ. The p-adic order or p-adic valuation for ℤ is defined as νp : ℤ → ℕ[2]
Rational numbers
The p-adic order can be extended into the rational numbers. We can define [3]
Some properties are:
Moreover, if , then
where inf is the infimum (i.e. the smaller of the two).
p-adic absolute value
The p-adic absolute value on ℚ is defined as |·|p : ℚ → ℝ
The p-adic absolute value satisfies the following properties:
Non-negativity Positive-definiteness Multiplicativity Subadditivity Non-Archimedean Symmetry
A metric space can be formed on the set ℚ with a (non-Archimedean, translation-invariant) metric defined by d : ℚ × ℚ → ℝ
The p-adic absolute value is sometimes referred to as the "p-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity.
See also
References
- ↑ Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
- ↑ Ireland, K.; Rosen, M. (2000). A Classical Introduction to Modern Number Theory. New York: Springer-Verlag. p. 3.
- ↑ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.