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$\Bbb Z[X]/(p, f(X))$ is integral domain. ($p$ is a prime number and $f(X)$ is an irreducible element of $\Bbb Z[X]$)

I want to show that if $a, b\notin (p, f(X)), ab\notin(p,f(X))$, but I don't know how to solve it...

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This is false.

As a counterexample let's use $p=5$ and $f(X)=X^2+1$. Denote $I=(p,f(X))$. Then neither $a=x-2$ nor $b=x+2$ belong to $I$, but their product $$ ab=x^2-4=(x^2+1)-5\in I. $$

Please check the question. May be the requirement is that $f(X)$ should be irreducible in $\Bbb{Z}_p[X]$? Then the claim would become true. See this thread for that variant.

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Jyrki Lahtonen
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