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Chinese remainder theorem

The Chinese remainder theorem is the name applied to a number of related results in abstract algebra and number theory.

Table of contents
1 Simultaneous congruences of integers
2 Statement for principal ideal domains
3 Statement for general rings

Simultaneous congruences of integers

The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao published in 1247, is a statement about simultaneous congruences (see modular arithmetic). Suppose n1, ..., nk are positive integers which are pairwise coprime (meaning gcd(ni, nj) = 1 whenever ij). Then, for any given integers a1, ..., ak, there exists an integer x solving the system of simultaneous congruences

xai (mod ni)    for i = 1...k     (1)
Furthermore, all solutions x to this system are congruent modulo the product n = n1...nk.

A solution x can be found as follows. For each i, the integers ni and n/ni are coprime, and using the extended Euclidean algorithm we can find integers r and s such that r ni + s n/ni = 1. If we set ei = s n/ni, then we have

ei ≡ 1 (mod ni)    and     ei ≡ 0 (mod nj)    for ji.
The number x = ∑i=1..k ai ei then solves the given system (1) of simultaneous congruences.

For example, consider the problem of finding an integer x such that

x ≡ 2 (mod 3)
x ≡ 3 (mod 4)
x ≡ 2 (mod 5)
Using the extended Euclidean algorithm for 3 and 4×5 = 20, we find (-13) × 3 + 2 × 20 = 1 (i.e. e1 = 40). Using the Euclidean algorithm for 4 and 3×5 = 15, we get (-11) × 4 + 3 × 15 = 1 (hence e2 = 45). Finally, using the Euclidean algorithm for 5 and 3×4 = 12, we get 5 × 5 + (-2) × 12 = 1 (meaning e3 = -24). A solution x is therefore 2 × 40 + 3 × 45 + 2 × (-24) = 167. All other solutions are congruent to 167 modulo 60, which means that they are all congruent to 47 modulo 60.

Note that some systems of the form (1) can be solved even if the numbers ni are not pairwise coprime. The precise criterion is as follows: a solution x exists if and only if aiaj (mod gcd(ni, nj)) for all i and j. All solutions x are congruent modulo the least common multiple of the ni.

Using the method of successive substitution can often yield solutions to simultaneous congruences, even when the moduli are not pairwise coprime.

Statement for principal ideal domains

For a principal ideal domain R the Chinese remainder theorem takes the following form: If u1, ..., uk are elements of R which are pairwise coprime, and u denotes the product, then the ring R/uR and the product ring R/u1R x ... x R/ukR are isomorphic via the isomorphism

f :     R/uR    -->  R/u1R x ... x R/ukR
      x mod uR  |-> ( (x mod u1R), ..., (x mod ukR) )

The inverse isomorphism can be constructed as follows. For each i, the elements ui and u/ui are coprime, and therefore there exist elements r and s in R with r ui + s u/ui = 1. Set ei = s u/ui. Then the map
g :  R/u1R x ... x R/ukR  -->    R/uR    
     ( (a1 mod u1R), ..., (ak mod ukR) )  |->  ∑i=1..k ai ei mod uR  

Statement for general rings

One of the most general forms of the Chinese remainder theorem can be formulated for rings and (two-sided) ideals. If R is a ring and I1, ..., Ik are ideals of R which are pairwise coprime (meaning that Ii + Ij = R whenever ij), then the product I of these ideals is equal to their intersection, and the ring R/I is isomorphic to the product ring R/I1 x R/I2 x ... x R/Ik via the isomorphism

f :     R/I    -->  R/I1 x ... x R/Ik
      x mod I  |-> ( (x mod I1), ..., (x mod Ik) )