p-adic valuation

For a prime p and nonzero rational number x, the p-adic valuation v_p(x) is the exponent of p in x, counting denominator factors negatively. It satisfies v_p(xy)=v_p(x)+v_p(y) and v_p(x+y)≥min(v_p(x),v_p(y)) when x+y≠0.

Prime factorization: The expression of an integer as a product of primes, unique up to the order of factors. The valuation extracts one prime's exponent from this factorization.
Divisibility: The relation in which an integer divides another when their quotient is an integer. For integers, v_p(x)≥k exactly when p^k divides x.
Hensel's lemma: A theorem that lifts approximate roots of polynomial equations modulo prime powers to roots in p-adic settings under suitable conditions. Valuation bounds quantify the approximation conditions that make root lifting possible.
Real absolute value: The usual absolute value on rational or real numbers, measuring distance from zero in the ordered number system. Unlike the p-adic absolute value, it makes numbers with large magnitude far from zero.
Valuation (mathematics): A function measuring the multiplicative size of elements in a field or ring through specified algebraic rules. The p-adic valuation is a discrete, prime-indexed example of this broader structure.
p-adic order: The largest integer k such that p^k divides a nonzero integer, often denoted ord_p. This integer-valued order extends from integers to rationals by subtracting denominator exponents.
Newton polygon: A polygonal construction from coefficient valuations used to infer root valuations and factorization data. Its slopes encode valuations of polynomial roots.
Archimedean absolute value: An absolute value satisfying the ordinary triangle inequality but not the stronger ultrametric inequality. The valuation-derived absolute value is non-Archimedean instead.
p-adic absolute value: The absolute value on rational numbers defined by |x|_p=p^{-v_p(x)} for nonzero x, with |0|_p=0. Negating the valuation and exponentiating converts it into a non-Archimedean absolute value.
Congruence (number theory): A relation stating that two integers differ by a multiple of a specified modulus. Valuation bounds translate directly into divisibility and congruence conditions modulo powers of p.

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