In which field does this equation lies

Question

It is well known that if $f(x)$ is irreducible then $$g(x) \bmod f(x) = g(\alpha)$$, where $\alpha$ is a root of $f(x)$. I am skeptical that can I write this equality $g(x) \bmod f(x) = g(\alpha)$ as on the left have side we have a polynomial and on the right hand side we have a field element.

Answer 1

By division $\,\ g = q f + r\ $ for $\ r = g\bmod f$

so $\,\color{#c00}{f(\alpha) = 0}\,\Rightarrow g(\alpha) = q(\alpha) \color{#c00}{f(\alpha)} + r(\alpha) = r(\alpha)$