A finite integral domain is a field

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

Proof

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

It follows that is a field.

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