If x is irreducible, is ϕ(x) irreducible? Prove or give counter-example.

**JJMC89** · April 10th 2011, 10:20 AM

Suppose that $\phi : R \rightarrow S$ is a surjective ring homomorphism. Suppose that $x \in R$ is an irreducible element. Is it true that $\phi(x)$ is also irreducible? Prove it or give a counter-example.

**Deveno** · April 10th 2011, 10:41 AM

R = Z[x], r = x + 4, S = Z, φ:R-->S given by φ(anx^n +.....+a1x + a0) = a0.

x+4 is irreducible in Z[x], but 4 is not irreducible in Z.

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If x is irreducible, is ϕ(x) irreducible? Prove or give counter-example.

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