Following is the problem 2.16 in The Math Problems Notebook:
Prove that if
, then we do not have any nontrivial solutions of the equation
where
are rational functions. Solutions of the form
where
is a rational function and
are complex numbers satisfying
, are called trivial.
This problem is analogous to the Fermat’s Last Theorem (FLT) which states that for ,
has no nontrivial integer solutions.
The solution of this problem involves proof by contradiction:
Since any rational solution yields a complex polynomial solution, by clearing the denominators, it is sufficient to assume that is a polynomial solution such that
is minimal among all polynomial solutions, where
.
Assume also that are relatively prime. Hence we have
, i.e.
. Now using the simple factorization identity involving the roots of unity, we get:
where with
.
Since , we have
for
. Since the ring of complex polynomials has unique facotrization property, we must have
, where
are polynomials satisfying
.
Now consider the factors . Note that, since
, these elements belong to the 2-dimensional vector space generated by
over
. Hence these three elements are linearly dependent, i.e. there exists a vanishing linear combination with complex coefficients (not all zero) in these three elements. Thus there exist
so that
. We then set
, and observe that
.
Moreover, the polynomials for
and
since
. Thus contradicting the minimality of
, i.e. the minimal (degree) solution
didn’t exist. Hence no solution exists.
The above argument fails for proving the non-existence of integer solutions since two coprime integers don’t form a 2-dimensional vector space over .