An old post by Nick Halloway at sci.math:
Someone mentioned to that an analog of FLT is true in as well, i.e. there are no relatively prime [non-constant] polynomials , and in with
This is true. You can show it by adapting the false proof of FLT.
It’s only necessary to show it for an odd prime and for . The proof is similar for to an odd prime, so I will just show it for odd primes.
Proof by induction on the sum of the degrees of and .
Suppose the sum of the degrees is . Then factors as
where is a primitive ‘th root of unity. These factors are all relatively prime polynomials of degree . But has repeated factors. So it won’t work for this case.
Now assume the sum of the degrees is . Again factor as
These factors are relatively prime. But has factors with multiplicity divisible by . So each factor must be a ‘th power. So we have
So , and
Since neither nor are 0, and and are relatively prime, this violates the induction hypothesis.
This also shows it for polynomials in , since polynomials that are relatively prime in are also relatively prime in .