A finite integral domain is a field

Let $𝐷$ be a finite Integral domain. Then $𝐷$ is a Field, i.e. every nonzero element of $𝐷$ is a unit. #m/thm/ring

Proof

Let $𝑎$ be a nonzero, non-unity element of $𝐷$ (if it is unity it is trivially a unit). Since $𝐷$ is finite, there must exist some $𝑖 + 1 < 𝑗$ such that $𝑎 𝑖 = 𝑎 𝑗$ . By cancellation it follows $𝑎 𝑖 - 𝑗 = 1$ and hence $𝑎 𝑎 𝑖 - 𝑗 - 1 = 1$ so $𝑎$ is a unit.

It follows that $ℤ 𝑝$ is a field.

#state/tidy | #lang/en | #SemBr