Remark

By formula (23) of VIII, p. 340, we have

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

for every $a\in A$. If the characteristic of the field K is equal to 0 or a prime number $p$ that does not divide $n$, then we can replace the reduced trace with the trace in Proposition 5. On the other hand, if the characteristic of K is a prime number that divides $n$, then we have Tr$_{A/K}(a) = 0$ for every $a\in A$.