# Chinese Remainder Theorem

Need to prove two parts and must follow the Chinese Remainder Theorem.

Let be polynomials with integer coefficients of the same degree d. Let be integers which are relatively prime in pairs (i.e., ( for i j). Use the Chinese Remainder Theorem to prove there exists a polynomial f(x) with integer coefficients and of degree d with

mod , mod ,..... mod

i.e., the coefficient of f(x) agree with the coefficients of mod .

Show that if all the are monic, then f(x) may also be chosen monic.

[Apply the Chinese Remainder Theorem in Z to each of the coefficients separately.]

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Suppose , , ...

#### Solution Summary

The Chinese Remainder Theorem is applied to monic functions.