If a subring $B$ of a field $F$ is closed with respect to multiplicative inverses, then $B$ is a field.
Fields are commutative rings with unity, and every nonzero element has an inverse. A subring is closed with respect to addition, multiplication, and negatives.
So $B$ is a subring that's also closed with respect to multiplicative inverses... To show that it's a field, don't I need to show that it's closed with respect to additive inverses? I'm really not sure what to do
Since $B$ is a subring, $B$ is a ring. Also, since multiplication in $F$ is commutative, multiplication in $B$ is commutative. So $B$ is a commutative ring. Finally, you just need to show every nonzero element of $B$ has a multiplicative inverse in $B$, and $B$ will be a field.