I want to prove a result and I think I have done it, but I am not sure whether it is correct or not. I am working on permutation polynomials over finite fields, where I want to prove a rational function $$f(x)= \frac{-x^7-x^6+x^4+x+1}{x^7+x^6+x^3-x-1}$$ as one over $\mu_{2^{m+1}}$ where $\mu_d$ is defined as $$\mu_d= \{x \in \bar{\mathbb F_{q}}: x^d=1 \}$$, $ \mathbb F_{q}$ is finite field with $q$ elements and $\bar{\mathbb F_{q}}$ is algebraic closure of $ \mathbb F_{q}$.
As we know to prove a function one, we usually equate $f(x)=f(y)$ and conclude that $x=y.$ Using the similar methodology, I equated $f(x)=f(y)$ i.e. $$\frac{-x^7-x^6+x^4+x+1}{x^7+x^6+x^3-x-1}= \frac{-y^7-y^6+y^4+y+1}{y^7+y^6+y^3-y-1}$$.
Both left-hand side and right-hand side are rational functions. Therefore, they must be equal in their lowest form. Thus,
\begin{equation} -x^7-x^6+x^4+x+1 =-y^7-y^6+y^4+y+1 \end{equation} \begin{equation} x^7+x^6+x^3-x-1 = y^7+y^6+y^3-y-1 \end{equation} I am not sure about this step; can I equate these numerators and denominators over $\mu_{2^{m+1}}$ to do my further calculations?