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 .*