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!