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 is Zero or Prime, $p$ is prime.