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Thread: Integral Domain

  1. #1
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    Integral Domain

    I am having troubles with the following problem:

    Show that there is an integral domain with exactly 4 elements.

    Any suggestion?
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  2. #2
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    Quote Originally Posted by ynn6871 View Post
    I am having troubles with the following problem:

    Show that there is an integral domain with exactly 4 elements.

    Any suggestion?
    Let $\displaystyle F = \mathbb{Z}_2[x]$ - the polynomials with coefficients in $\displaystyle \mathbb{Z}_2$.
    Let $\displaystyle f(x) = x^2 + x + 1$. This polynomial is irreducible over $\displaystyle \mathbb{Z}_2[x]$.
    Therefore, all elements in $\displaystyle D=\mathbb{Z}_2[x]/(x^2+x+1)$ are invertible.
    Consequently, $\displaystyle D$ is an integral domain with $\displaystyle |D| = 4$.
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