Definition 1

Let $M$ be an $A$-module admitting a finite basis as a $K$-module. For all $a \in A$, let $a_M$ be the endomorphism $x \mapsto ax$ of the $K$-module $M$. The trace, determinant and characteristic polynomial of $a_M$ are called respectively the trace, norm and characteristic polynomial of $z$ relative to $M$.

The trace and norm of $a$ are therefore elements of $K$, denoted respectively by $\operatorname{Tr}_{M/K}(a)$ and $N_{M/K}(a)$; the characteristic polynomial of $a$ is an element of $K[X]$, denoted by $\mathrm{Pc}_{M/K}(a; X)$. We omit $K$ in the above notation when there is no risk of confusion.

From the properties of the trace and determinant of an endomorphism (II, § 4, no. 3 and § 8, no. 1) we obtain the relations

(2) $\mathrm{Tr}_M(aa') = \mathrm{Tr}_M(a'a)$ (3) $N_M(aa') = N_M(a)N_M(a')$

for all $a, a'$ in $A$.

Let $(e_i)_{1 \leq i \leq n}$ be a basis of the $K$-module $M$ and $(m_{ij}(a))$ the matrix of the endomorphism $a_M$ with respect to this basis. The functions $m_{ij}$ are linear forms on the $K$-module $A$ and

(4) $\mathrm{Tr}_M(a) = \sum_{i=1}^n m_{ii}(a)$ (5) $N_M(a) = \det(m_{ij}(a))$ (6) $\mathrm{Pc}(a; X) = \det(\delta_{ij}X - m_{ij}(a)).$

It follows from the method of calculating a determinant (§ 8, no. 11, formula (50)) that

(7) $\mathrm{Pc}_M(a; X) = X^n + c_1 X^{n-1} + \cdots + c_n$

where

(8) $c_1 = -\mathrm{Tr}_M(a), \quad c_n = (-1)^n N_M(a).$

For $\lambda \in K$,

(9) $\mathrm{Tr}_M(\lambda) = n.\lambda, \quad N_M(\lambda) = \lambda^n, \quad \mathrm{Pc}_M(\lambda; X) = (X - \lambda)^n.$

Let $K'$ be a commutative $K$-algebra. Let $M' = K' \otimes_K M$ and $A' = K' \otimes_K A$, so that $M'$ has an $A'$-module structure (§ 4, Example 2). As a $K'$-module $M'$ admits the basis consisting of the $1 \otimes e_i$ ($1 \leq i \leq n$) and the matrix of $a_M$ with respect to $(e_i)$ is equal to the matrix of $(1 \otimes a)_M'$ with respect to $(e'_i)$. Then

(12) $\mathrm{Tr}_{M'}(1 \otimes a) = \mathrm{Tr}_M(a).1, \quad N_{M'}(1 \otimes a) = N_M(a).1$ $\mathrm{Pc}_{M'}(1 \otimes a; X) = \mathrm{Pc}_M(a; X).1$

for all $a \in A$, where 1 denotes the unit element of $K'$. If in particular we take $K' = K[X]$, then

(13) $\mathrm{Pc}_{M/K}(a; X) = N_{M[X]/K[X]}(X - a).$