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Thread: Probably a simple explanation

  1. #1
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    Probably a simple explanation

    Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
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  2. #2
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    Quote Originally Posted by Ryaη View Post
    Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
    It factorises as 2(x5) in both cases. However, 2 is a unit in \mathbb{Q}[x] but not in \mathbb{Z}[x].
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    Quote Originally Posted by Opalg View Post
    It factorises as 2(x5) in both cases. However, 2 is a unit in \mathbb{Q}[x] but not in \mathbb{Z}[x].
    Why would that not be the other way around? (since it is irreducible in Q[x] and reducible in Z[x])
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  4. #4
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    Quote Originally Posted by Ryaη View Post
    Why would that not be the other way around? (since it is irreducible in Q[x] and reducible in Z[x])
    Reducible means having a nontrivial factorisation. Irreducible means having no nontrivial factorisation. Nontrivial means that neither of the factors is a unit. If one of the factors is a unit then the factorisation doesn't count as a nontrivial factorisation. Since 2 is a unit in \mathbb{Q}, the factorisation 2x – 10 = 2(x – 5) doesn't count as nontrivial. That is why 2x – 10 is irreducible in \mathbb{Q}[x].
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