Definition 2

Let $A$ be a $K$-algebra which is a finite-dimensional free $K$-module. For every element $a \in A$, the trace (resp. norm,\footnote{† This notion should not be confused with the notion for norm in an algebra over a valued field (\emph{General Topology}), IX, § 3, no. 7.} resp. characteristic polynomial) of $a$ relative to $A$ and $K$ is the trace (resp. determinant, resp. characteristic polynomial) of the endomorphism $x \mapsto ax$ of the $K$-module $A$.

The trace, norm and characteristic polynomial of $a \in A$ relative to $A$ and $K$ are denoted by $\mathrm{Tr}_{A/K}(a)$, $N_{A/K}(a)$ and $\mathrm{Pc}_{A/K}(a; X)$; we omit $K$ and even $A$ from this notation when there is no risk of confusion. Note that the trace (resp. norm, characteristic polynomial) of $a \in A$ is just the trace (resp. norm, characteristic polynomial) of $a$ relative to the $A$-module $A_s$.

Suppose that $A$ is the product $A_1 \times A_2 \times \cdots \times A_m$ of a finite number finite-dimensional algebras over $K$ which are free $K$-modules. Using the above remark and Proposition 1 of no. 2, we have, for every element

(18)

Similarly, Proposition 2 of no. 2 shows that if $A$ and $A'$ are two algebras, which are free $K$-modules, of finite dimensions $n, n'$ respectively over $K$, then, for $a \in A, a' \in A'$.

(19)

(20)

Finally let $A$ be a finite-dimensional algebra over $K$ which is a free $K$-module, $h$ a homomorphism of $K$ into a commutative ring $K'$ and $A' = A_{(K')}$ the $K'$-algebra derived from $A$ by extending the scalars by means of $h$. It follows from formula (12) of no. 1 that, for all $a \in A$,

(21)

where $\bar{h}$ is the homomorphism $K[X] \to K'[X]$ derived from $h$. In particular, for $K' = K[X]$, we obtain, writing $A[X] = A \otimes_K K[X]$,

(22)

More generally, if $K'$ is a commutative $K$-algebra and $A' = A \otimes_K K'$, then, for all $x \in A'$,