Corollary

Let D be a field of finite degree over K with center K. Let $a$ be an element of K and L a maximal commutative subfield of D containing $a$. We have

Pcrd$_{D/K}(a; X) =$ Pc$_{L/K}(a; X)$,

(55) Tr$_{D/K}(a) =$ Tr$_{L/K}(a)$,

Nrd$_{D/K}(a) = N_{L/K}(a)$.

Indeed, a maximal commutative subfield L of D is a maximal commutative semisimple subalgebra of D by Corollary 2 of VIII, p. 265.