Is $F=\{-1, 0, 1\}$ a subfield of $\mathbb C$?
I have just started reading Linear Algebra text by Kenneth Hoffman and Ray Kunze. They wrote that "any subfield of $\mathbb C$ must contain every rational number", this statement implies that $F$ is not a subfield of $\mathbb C$ even though the operations of addition and multiplication on $F$ are defined in $F$.
Am I overlooking some fact?
$F$ is not a subfield of $\mathbb{C}$; it is not closed under addition.
Example: $1+1=2\notin F$. Although $1$ is an element of $F$, $2$ is not.