# Finite-Dimensional Integral Domain over Field is Field

## Theorem

Let $k$ be a field.

Let $R$ be a $k$-algebra of finite dimension which is an integral domain.

Then $R$ is a field.

## Proof

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Let $k$ be a field.

Let $R$ be a $k$-algebra of finite dimension which is an integral domain.

Then $R$ is a field.

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