be the Jordan-Chevalley decomposition of .
Let be the minimal polynomial of . If all of the roots of are separable over , then and are defined over . (That means, they are matrices over ).
Let us define a polynomial as follows.
These polynomials are designed to satisfy the following property.
Then we further define
and
It is fairly easy to see that
holds.
The function is symmetric with respect to roots and thus is a polynomial with coefficients in . Thus (hence also ) is defined over .
The following example shows that the -rationality of does not necessarily hold when we drop off the assumption on .
Then the minimal polynomial of is given by . The Jordan-Chevalley decomposition of is given by
Thus the decomposition is not defined over .