# Prove that $B$ is a subfield of $F$

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

• I know. But that doesn't mean it's necessarily a field? Commented May 7, 2014 at 3:26
• If $B$ is a (sub)ring, it’s closed under $+$, $-$, and $\cdot$. Commented May 7, 2014 at 3:28
• So that implies that it is closed with respect to additive inverses? ...since $a, -a \in B$, $a+(-a)=a-a=0$? So $-a$ is the additive inverse for $a$? Commented May 7, 2014 at 3:36
• What does being "closed wrt multiplicative inverses" mean? Does this mean that any non zero element in $\;B\;$ has an inverse in $\;B\;$ ? Because this much is all that needs to be proved in order to get a field, as $\;B\;$ is already an integral domain... Commented May 7, 2014 at 3:57
• Yes, being closed under subtraction is the same as having additive inverses. Can you see that? Maybe you didn’t catch on that “additive inverses” and “negatives” are just different words for the same thing? Commented May 7, 2014 at 16:28

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.