Properties of Prime Subfield

Theorem
Let $$F$$ be a field.

Let $$K$$ be the prime subfield of $$F$$.

Then $$K$$ is isomorphic to either:


 * $$\Q$$, the Field of Rational Numbers, or


 * $$\Z_p$$, the Ring of Integers Modulo $p$, where $$p$$ is prime.

Proof
From Field of Characteristic Zero has Unique Prime Subfield, if $$\operatorname{Char} \left({F}\right) = 0$$, then its prime subfield is isomorphic to $$\Q$$, the Field of Rational Numbers.

From Field of Prime Characteristic has Unique Prime Subfield, if $$\operatorname{Char} \left({F}\right) = p$$, then its prime subfield is isomorphic to $$\Z_p$$, the Ring of Integers Modulo $p$.

From Characteristic of Field, $$p$$ is prime.