Properties of Prime Subfield

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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 $\Char F = 0$, then its prime subfield is isomorphic to $\Q$, the field of rational numbers.

From Field of Prime Characteristic has Unique Prime Subfield, if $\Char F = p$, 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.

$\blacksquare$


Sources