Finite Integral Domain is Galois Field/Proof 4
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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 $7 \ \text {(ii)}$