Editing P-adic number (section)

== Modular properties ==
The [[quotient ring]] <math>\Z_p/p^n\Z_p</math> may be identified with the [[ring (mathematics)|ring]] <math>\Z/p^n\Z</math> of the integers [[modular arithmetic|modulo]] <math>p^n.</math> This can be shown by remarking that every {{mvar|p}}-adic integer, represented by its normalized {{mvar|p}}-adic series, is congruent modulo <math>p^n</math> with its [[partial sum]] <math display = inline>\sum_{i=0}^{n-1}a_ip^i,</math> whose value is an integer in the interval <math>[0,p^n-1].</math> A straightforward verification shows that this defines a [[ring isomorphism]] from <math>\Z_p/p^n\Z_p</math> to <math>\Z/p^n\Z.</math>

The [[inverse limit]] of the rings <math>\Z_p/p^n\Z_p</math> is defined as the ring formed by the sequences <math>a_0, a_1, \ldots</math> such that <math>a_i \in \Z/p^i \Z</math> and <math display = inline>a_i \equiv a_{i+1} \pmod {p^i}</math> for every {{mvar|i}}.

The mapping that maps a normalized {{mvar|p}}-adic series to the sequence of its partial sums is a ring isomorphism from <math>\Z_p</math> to the inverse limit of the <math>\Z_p/p^n\Z_p.</math> This provides another way for defining {{mvar|p}}-adic integers ([[up to]] an isomorphism).

This definition of {{mvar|p}}-adic integers is specially useful for practical computations, as allowing building {{mvar|p}}-adic integers by successive approximations.

For example, for computing the {{mvar|p}}-adic (multiplicative) inverse of an integer, one can use [[Newton's method]], starting from the inverse modulo {{mvar|p}}; then, each Newton step computes the inverse modulo <math display = inline>p^{n^2}</math> from the inverse modulo <math display = inline>p^n.</math>

The same method can be used for computing the {{mvar|p}}-adic [[square root]] of an integer that is a [[quadratic residue]] modulo {{mvar|p}}. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in <math>\Z_p/p^n\Z_p</math>. Applying Newton's method to find the square root requires <math display = inline>p^n</math> to be larger than twice the given integer, which is quickly satisfied.

[[Hensel lifting]] is a similar method that allows to "lift" the factorization modulo {{mvar|p}} of a polynomial with integer coefficients to a factorization modulo <math display = inline>p^n</math> for large values of {{mvar|n}}. This is commonly used by [[polynomial factorization]] algorithms.