I am trying to make a program that, among other things, considers a polynomial $p$ whose coefficients are elements of $\mathrm{GF}\left(2^{8}\right)$ and shows the user the graph of that polynomial. This necessarily implies that I must calculate the image of some numbers $\alpha_{1}$, $\alpha_{2}$, $\ldots$, $\alpha_{n - 1}$ and $\alpha_{n}$ under $p$, and this is where my confusion starts: to which set must $\alpha_{k}$ belong (with $k \in \left[1, n\right] \cap \mathbb{Z}$)?
In order not to have problems when performing the calculations, I could consider that all these $\alpha_{k}$ belong to the set $\mathrm{GF}\left(2^{8}\right)$, since, in this case, expressions like $$ \left[00000111\right]_2 \alpha_{1}^{2} + \left[00001010\right]_2 \alpha_{1} + \left[00000001\right]_2$$ would be well defined. But does it necessarily have to be so? Can't it be possible that $\alpha_{k}$ is some element of $\mathbb{R}$? And if it is possible, how should sum and product be defined so that expressions like the one I wrote make sense for values like $0.5$?
I apologize in advance if my doubt is nonsense. I have very little knowledge about finite fields, as all I know about them is due to the program I am trying to do, and, it is funny, but in all the files and videos in which I consulted about this question, nobody explains how the heck to perform the evaluation of a polynomial that belongs to $\mathrm{GF}\left(2^{8}\right)\left[x\right]$.
I deeply appreciate any answers you can offer me.