Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent.
Problem Statement
Suppose we have rational functions $f,g \in \mathbb C[t]$. We want to prove that these have degree 0 if $f^4(t)+g^4(t) = 1$, and we want to do so utilizing infinite descent.
note: I know how to do it for the $n=3$ case via infinite descent utilizing the roots of unity, but I'm having trouble applying the same workings to this example.
Current working so far
Let $f = \frac pr$ and $g = \frac qr$ for co-prime $p,q,r \in \mathbb C[t]$. Then plugging in and rearranging gives $$p^4(t)+q^4(t)=r^4(t).$$ From here, I rearrange and expand the above: $$(p(t)+r(t))(p(t)-r(t))(p(t)+ir(t))(p(t)-ir(t)) = -q^4(t).$$
My idea from here is to set new polynomials $\alpha,\beta,\gamma$ s.t. one of the parts of the above product is equal to a corresponding polynomial to the 4th power, claiming that the degree of one of $\alpha,\beta,\gamma$ is one fourth the degree of one of $p,q,r$. If that were the case, then I could use proof by infinite descent to then prove the claim.
Query
However, after playing around with the terms, I'm quite certain it's not doable, at least not how I'm doing it. Am I on the right track, or do I need to rethink how I go about this? (Any hints would be appreciated!)