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Thread: Irreducible polynomial ring

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

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

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

    Suppose $\displaystyle f$ is irreducible.

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

    WLOG, take $\displaystyle b=1$.

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

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

    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 $\displaystyle f(x)=g(x)h(x)$, then $\displaystyle f_1(x)=f(x+1)=g(x+1)h(x+1)=g_1(x)h_1(x)$.
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