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Math Help - Irreducible polynomial ring

  1. #1
    Super Member Showcase_22's Avatar
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    Irreducible polynomial ring

    If f \in \mathbb{Z}[x], then define f_1 \in \mathbb{Z}[x] by f_1(x)=f(x+1).

    Show that f is irreducible iff f_1 is irreducible.
    I tried doing it this way:

    Suppose f is irreducible.

    Then f(x)=(a_0+a_1x^1+ \ldots +a_nx^n).b where b is a unit (ie. b= \pm 1).

    WLOG, take b=1.

    Therefore f_1(x)=f(x+1)=a_0+a_1(x+1)+ \ldots + a_n(x+1)^n

    = \sum_{i=0}^n a_i+ \left( \sum_{i=1}^n a_i \right).x+ \ldots + a_nx^n

    From here i'm not really sure where to go. Can anyone help?
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  2. #2
    Senior Member Tinyboss's Avatar
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    If f(x)=g(x)h(x), then f_1(x)=f(x+1)=g(x+1)h(x+1)=g_1(x)h_1(x).
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