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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 61.3$ Characteristic of an integral domain or field