A Universe of Sorts

Take $Q(\sqrt 2)$ over $Q$ . $\sqrt(2)$ corresponds to the linear transform $[0 1][2 0]$ over the basis $a + b \sqrt 2$ .
The chracteristic polynomial of the linear transform is $x^2 - 2$ , which is indeed the minimal polynomial for $\sqrt(2)$ .
Asking for every element of $Q(\sqrt 2)$ to be separable is the same as asking every element of $Q(\sqrt 2)$ interpreted as a linear opearator to have separable minimal polynomial.
Recall that the minimal polynomial is the lowest degree polynomial that annhilates the linear operator. So $minpoly(I) = x - 1$ , $charpoly(I) = (x - 1)^n$ .

§ Separability of Field Extension as Diagonalizability