Chinese remainder theorem
Let be a commutative ring with identity and let
be a collection of ideals in
such that
for any
. Then, given any set of elements
, there exists an
such that
,
. In the particular case when
is the ring of integers
, the Chinese remainder theorem states that for any set of pairwise coprime numbers
there is an integer
giving pre-assigned remainders on division by
. In this form the Chinese remainder theorem was known in ancient China; whence the name of the theorem.
The most frequent application of the Chinese remainder theorem is in the case when is a Dedekind ring and
, where the
are distinct prime ideals in
. (If
satisfy the condition of the theorem, then so do
for any natural numbers
.) In this case, the Chinese remainder theorem implies that for any set
there exists an
such that the decomposition of the principal ideal
into a product of prime ideals has the form
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where the ideals are pairwise distinct (the theorem on the independence of exponents).
References
[1] | A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) |
[2] | S. Lang, "Algebra" , Addison-Wesley (1974) |
[3] | S. Lang, "Algebraic numbers" , Addison-Wesley (1964) |
Chinese remainder theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chinese_remainder_theorem&oldid=39089