1. ## Subfield Q

Show that K is a subfield of F if and only if $0\neq a\in K$ implies that $a^{-1}\in K$

So, we know that all fields have inverses. So, if $0\neq a$ then we can say $a=n\in K$. Then we have $n+(-n)=0$ and $nn^{-1}=1=n^{-1}n$ because $a\neq 0$. Now this doesn't completely show $\rightarrow$ part of the iff does it? If we know that the element has an inverse, can we say it satisfies all of the other axioms of a field?

2. Show that K is a subfield of F if and only if implies that
Arn't there other hypotheses? I mean, if that equivalence was true, then $(\{0,1\},+,\times )$ would be a subfield of $(\mathbb{Q},+,\times ),$ and this is obviously wrong.

A subfield is a subring, so it is an additive subgroup.

3. no other hypotheses, just what i had written. Interesting. But why is 0 in the first ring when a cannot equal 0?

4. Ah, i believe this proof is quite trivial. Since a subfield is a subgring, all we need to say is that in additionto the things required to be a subfield we only need to show it also has an inverse.