I am fully aware I may be asking a silly question here.
But surely, polynomials with unity as coefficients is well defined over any field since every field contains 1.
So does the separability, roots change depending on the base field?
Say if I had a based field with char 0, or $p_1$, or $p_2$, what difference does it make?
thank you for enlightening me.
Well to begin, every field contains some NOTION of $1$, but they will not always coincide. For instance, the $"1"$ in $Z_p$ is actually $1 + p\mathbb{Z}$.
So for polynomials in general you need a specification as to what ring its coefficients are in.
As to specifying what field the roots are in, it's still permissible to leave out what extension you are talking about; it is then understood to mean you are talking about its roots in the polynomial's splitting field.
The reason it matters in general is best seen by an example:
Consider the polynomial $x^2+1 \in \mathbb{R}[x]$. This polynomial has no roots in $\mathbb{R}$. However, it does have roots in $\mathbb{C}$, namingly $\pm i$.
This distinction matters, for instance, when you want to talk about normal extensions. An extension $E/F$ is normal if any irreducible $g \in F[x]$ having a root in $E$ splits in $E$ (all of its roots in E). So you can see in this example, the specific extension the roots are in matters. These extensions are of central interest in Galois Theory.