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Math Help - X^2 injective in a ring

  1. #1
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    X^2 injective in a ring

    Let A be a finite comutative ring . If f:A->A , f(x)=X^2 is injective prove that :
    a) 1+1=0
    b) P(x)=x^{2n}+a is reductible for all n \in N and for all  a \in A
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  2. #2
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    Convince yourself that f(-1)=f(1) and it follows (because f is injective) that -1=1 or equivalently 1+1=0.
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  3. #3
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    Part b) is a bit trickier.

    f is injective so there exists b \in A such that f(b)=a. Hence a^{1/2} is exists.

    Part a) shows that A has characteristic 2 and hence
    P(x)=x^{2n}+a=(x^n+a^{1/2})^2

    Ta da. P(x) is reducible
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