§ Separability of field extension as diagonalizability
- Take over . corresponds to the linear transform over the basis .
- The chracteristic polynomial of the linear transform is , which is indeed the minimal polynomial for .
- Asking for every element of to be separable is the same as asking every element of 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 , .