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Thread: Commutative ring

  1. #1
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    Commutative ring

    Let $\displaystyle R$ be a commutative ring with $\displaystyle r \in R$.
    Let $\displaystyle f(x) $ be an element of $\displaystyle R[x]$.
    I want to show that $\displaystyle f(x)$ is irreducible over $\displaystyle R$ if and only if $\displaystyle f(x + r) $ irreducible over $\displaystyle R$.
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  2. #2
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    Quote Originally Posted by bram kierkels View Post
    Let $\displaystyle R$ be a commutative ring with $\displaystyle r \in R$.
    Let $\displaystyle f(x) $ be an element of $\displaystyle R[x]$.
    I want to show that $\displaystyle f(x)$ is irreducible over $\displaystyle R$ if and only if $\displaystyle f(x + r) $ irreducible over $\displaystyle R$.


    The map $\displaystyle p(x)\mapsto p(x+r),\,r\in R$ is an automorphism of the ring $\displaystyle R[x]$, and then:

    $\displaystyle f(x)$reducible $\displaystyle \Longleftrightarrow f(x)=g(x)q(x),\,g(x),\,q(x)\in R[x]\Longleftrightarrow f(x+r)=g(x+r)q(x+r)$ via the above automorphism $\displaystyle \Longleftrightarrow f(x+r)$ reducible .

    Tonio
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