Proposition 4

Let L be an extension of K and $a$ an element of the central simple algebra A. We have the relations

(28) Pcrd$_{A_{(L)}/L}(1\otimes a; X) =$ Pcrd$_{A/K}(a; X)$,

(29) Trd$_{A_{(L)}/L}(1\otimes a) =$ Trd$_{A/K}(a)$,

(30) Nrd$_{A_{(L)}/L}(1\otimes a) =$ Nrd$_{A/K}(a)$

(“invariance under extension of scalars”).

The two sides of equality (28) have the same $n$-th power by the definition (formula (20) of VIII, p. 340) and the relation

Pc$_{A_{(L)}/L}(1\otimes a; X) =$ Pc$_{A/K}(a; X)$ (III, §9, No. 3, p. 544, formula (21)). Equality (28) therefore follows from Lemma 2 of VIII, p. 339. Formulas (29) and (30) follow from (28), (21), and (22).