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Math Help - Wrong(?) Question about Irreducibiliity of Polynomials

  1. #1
    Senior Member slevvio's Avatar
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    Wrong(?) Question about Irreducibiliity of Polynomials

    Here is a question from my Galois Theory course:

    Let f be a polynomial over \mathbb{Z} with degree 2 or 3, and suppose it is reducible over \mathbb{Q}. What is wrong with the following argument: f is reducible over \mathbb{Z} by Gauss's Lemma and so must have a zero over \mathbb{Z} [Give both a counterexample and show where the error in the proof is].

    But wait a minute! Isn't the statement above true by the contrapositive of Gauss's Lemma (i.e. f irreducible over Z => f irreducible over Q)? And isn't it true anyway since the ring of integers of Q is Z, and hence by definition since f has coefficients in Z and is factorisable, its roots will lie in the algebraic integers, which are the integers ? Thanks if anyone could help me with this!
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  2. #2
    MHF Contributor

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    what if f(x) = 4x^2 + 12x + 9?

    clearly f(x) factors over Z as (2x + 3)^2

    are the roots to this equation in Z?
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  3. #3
    Senior Member slevvio's Avatar
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    Sorry - Gauss' Lemma doesn't tell you about the roots! oops
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