Finite Integral Domain is Galois Field/Proof 4

Theorem

A finite integral domain is a Galois field.

Proof

Aiming for a contradiction, suppose $\struct {D, +, \circ}$ is a finite integral domain which is not a field.

From Non-Field Integral Domain has Infinite Number of Ideals, $\struct {D, +, \circ}$ has an infinite number of distinct ideals.

But this contradicts the assertion that $\struct {D, +, \circ}$ is finite.

Hence the result by Proof by Contradiction.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $7 \ \text {(ii)}$