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Show that every finite integral domain is a field.

I have shown that every nonzero element in a finite ring with an identity is either a zero divisor or a unit.

Jyrki Lahtonen
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Moe
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1 Answers1

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Take $x\in R^*$. For any $k\in\mathbb{Z}$ $x^k\neq0$, because $R$ is integral domain. But $|R|=n$, $|R^*|=n-1$, so $|\{x^1,..,x^n\}|<n$. There exists $a,b\in \{1,,n\},\ a<b$ such that $x^a=x^b$, thus $x^{b-a}=1$ and $x$ is invertible because $ x^{-1}=x^{b-a-1}$. If every nonzero element is invertible, then $R$ is a field.

Przemek
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