Field

A field contains or

Let be a field. Then has a unique subfield isomorphic to modular arithmetic for some prime or . #m/thm/ring Thus and are the only prime fields.

Proof

The characteristic of an integral domain is 0 or prime, and A ring contains or where . This covers the case. If , then let be the unique initial ring homomorphism and

Then .

Thus has initial iff and initial if is prime.

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