Lemma 5

For every extension $K'$ of K and every element $(t_i)_{i\in I}$ of ${K'}^I$, we have

(48) Pcrd$_{A_{(K')}/K'}(\sum_{i\in I}t_i\otimes e_i)= P((t_i)_{i\in I},X)$.

Let $\varphi : K[\mathbf{T}]\rightarrow K'$ be the unique K-algebra homomorphism that sends $T_i$ to $t_i$ for every $i\in I$; it defines on $K'$ the structure of a $K[\mathbf{T}$]-algebra. The $K'$-algebra $A_{(K[\mathbf{T}])(K')}$ can be identified with $A_{(K')}$ (transitivity of extension of scalars), where the element $1\otimes (\sum T_i\otimes e_i)$ is identified with the element $\sum t_i\otimes e_i$ of $A_{(K')}$. We denote by $\overline{\varphi}: K[\mathbf{T}][X]\rightarrow K'[X]$ the K-algebra homomorphism deduced from $\varphi$. By formula (21) of III, §9, No. 3, p. 544, the characteristic polynomial of $\sum t_i\otimes e_i$ with respect to the $K'$-algebra $A_{(K')}$ is the image by $\overline{\varphi}$ of the characteristic polynomial of $\sum T_i\otimes e_i$ with respect to the $K[\mathbf{T}$]-algebra $A_{(K[\mathbf{T}])}$, that is, of $P^n$. In other words, we have

(49) Pc$_{A_{(K')}/K'}(\sum_{i\in I}t_i\otimes e_i; X)= P((t_i)_{i\in I},X)^n$;

Lemma 5 then follows from Lemma 2 of VIII, p. 339.

Consider the specific case $K'= K$ of Lemma 5. We have

(50) Pcrd$_{A/K}(\sum_{i\in I}t_ie_i; X)= P((t_i)_{i\in I},X)$

for every element $(t_i)_{i\in I}$ of $K^I$. Since the polynomial P in $K[\mathbf{T},X]$ is homogeneous of degree $n$, it can be expanded uniquely as

where $B_r$ is a homogeneous polynomial of degree $r$ in $K[\mathbf{T}]$. By formulas (46), (50), and (51), we have

for every element $(t_i)_{i\in I}$ of $K^I$. Proposition 6 follows.