Definition 2

We call Trd$_{A/K}(a)$ the reduced trace of A and Nrd$_{A/K}(a)$ its reduced norm (with respect to the K-algebra A).

When there is no risk of confusion, we leave A and K out of the notation and simply write Pcrd($a; X$), Trd($a$), and Nrd($a$).

The following formulas result from formulas (20) and (22) and formulas (7) and (8) of III, §9, No. 1, p. 542:

(23) Tr$_{A/K}(a) =n$ Trd$_{A/K}(a)$,

(24) $N_{A/K}(a) =$ (Nrd$_{A/K}(a)$)$^n$.