§ Separability of field extension as diagonalizability

• 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 $minpoly(I) = x - 1$, $charpoly(I) = (x - 1)^n$.