Definition 1. Let $(F, +, \cdot)$ be a field. $(K, +, \cdot)$ is a subfield of $F$ if it is a field itself and $K \subseteq F$.
Proposition 1 (Subfield Test). Let $F$ be a field. $K$ is a subfield of $F$ if:
-
$K$ is a subring.
-
(Closure under inversion) For all $a \in K \setdif \{0\}$,
\[a^{-1} \in K.\]