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

  1. #1
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    If x is irreducible, is ϕ(x) irreducible? Prove or give counter-example.

    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.
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  2. #2
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    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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