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Thread: Rings and Integral Domains

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    Super Member Aryth's Avatar
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    Rings and Integral Domains

    Let $\displaystyle \mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\displaystyle \mathcal{Z}$ and the usual operations of addition and multiplication.

    Let $\displaystyle S$ be the set of polynomials in $\displaystyle \mathcal{Z}[x]$ with roots at $\displaystyle x=0$ and $\displaystyle x=1$. Prove that $\displaystyle S$ is a subring of $\displaystyle \mathcal{Z}[x]$.
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    Quote Originally Posted by Aryth View Post
    Let $\displaystyle \mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\displaystyle \mathcal{Z}$ and the usual operations of addition and multiplication.

    Let $\displaystyle S$ be the set of polynomials in $\displaystyle \mathcal{Z}[x]$ with roots at $\displaystyle x=0$ and $\displaystyle x=1$. Prove that $\displaystyle S$ is a subring of $\displaystyle \mathcal{Z}[x]$.
    Do it by definition.
    For example, for additive closure say $\displaystyle f(x),g(x) \in S$.
    Then $\displaystyle f(x)+g(x) \in S$ since $\displaystyle f(0)+g(0) = 0 + 0 = 0$ and $\displaystyle f(1)+g(1) = 0 + 0 = 0$.
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