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Math Help - Commutative ring

  1. #1
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    Commutative ring

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


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

    f(x)reducible \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 \Longleftrightarrow f(x+r) reducible .

    Tonio
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