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I wanted to know something regarding the quotient ring Z[x]/pZ[x], where Z[x] is the set of all polynomials with integer coefficients and pZ[x], for a prime number p, is the set of all polynomials with integer coefficients divisible by p. I'm currently working through some notes on Applying Basic Abstract Algebra to Problems of Number Theory, and I don't understand the proof of Eisenstein's criterion. My confusion hinges upon this issue. The author of the notes says that if p(x) is a reducible polynomial with integer coefficients, then p(x)+Z[x]={q(x)+Z[x]}{r(x)+Z[x]}, where q(x) and r(x) are elements of Z[x] such that q(x)r(x)=p(x). Now, I know this seems really basic, but I have a question: Doesn't this hold for general integers rather than prime numbers? Must p be prime?

Thanks,

Mathguy

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# Ring theory question

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