Assume I am given two polynomials and with coefficients from a field , where is a prime. Now I know that the set of these polynomials is a ring and not a field, meaning that not every polynomial has an inverse.
Given evaluation points for some distinct values , one can first compute evaluation points and then interpolate the polynomial from them such that the interpolated polynomial contains the correct roots in the numerator and denominator. For example if the polynomial has roots ,, and has roots , , and , then the interpolated rational function will have a numerator with the root and the denominator will have the root .
My confusion stems from the fact that may not have an inverse, yet we can interpolate a polynomial , which is basically an inverse or not?
P.S. In case it helps, I only care about polynomials of the form , where for all it holds that .