I think I made some progress? not sure if it is valid...

Since

, then we can write each element

as

for some

.

But since

itself is in

, then

for some

.

Since

is commutative,

Does this mean that we have a unity? How do we prove that

works for every element of

?

Once we can show we have a unity, I think the rest is done, because:

if

, then

. So each element of

can be written as

for some

. Since

,

for some

. Since

is commutative,

, so by the definition of unique inverses, each element has an inverse, so each element is a unit, so

is a field.

Can anyone look over this argument and verify it/spot a problem? I'd be very grateful, thanks!