This is one of the elegant results about fields.

Theorem:If is a ring with unity and with charachteristic , then contains as a subring up to isomorphism.

Thus, given a field with a charachetristic then must be a prime. Because otherwise it will have zero-divisors, which a field does not have. Thus it contains as a subfield up to isomorphism.

This is a known fact that is always a field. Because it is a finite integral domain, and a finite integral domain is always a field.