Question

In: Advanced Math

Formulate the Chinese Remainder Theorem for polynomials over Z/p and prove it. (Please don't just copy...

Formulate the Chinese Remainder Theorem for polynomials over Z/p and prove it. (Please don't just copy and paste another solution onto here)

Expert Solution

Let and be positive integers which are relatively prime and let and be any two integers. Then there is an integer such that

N EQ a(mod r) (1)

and

N EQ b(mod s) (2)

Moreover, N is uniquely determined modulo r s.

An equivalent statement is that if , then every pair of residue classes modulo r and s corresponds to a simple residue class modulo .

The Chinese remainder theorem is implemented in the Wolfram Language as ChineseRemainder[a1, a2, ...m1, m2, ...]. The Chinese remainder theorem is also implemented indirectly using Reduce in with a domain specification of Integers.

The theorem can also be generalized as follows. Given a set of simultaneous congruences

(3)

for , ..., and for which the are pairwise relatively prime, the solution of the set of congruences is

(4)

where

(5)

and the are determined from

Colby Messinger answered 2 years ago

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