Given we attempt to find a such as,Originally Posted by suedenation
because 2 is the identity element.
by definition on .
or more elegantly
Now we examine 2 cases:
Case 1 : Thus, . Impossible.
Case 2 : Thus, . Thus if then Checking this we find that which is true. Thus, has an inverse. Now if also then,
with because they are right next to each other. Thus, the left hand side must be divisible by , thus, this is Euclid's Lemma. Thus, . Thus, now since thus, happens only when . Thus,
Thus, is the other inverse. Thus, are the only possible inverses.