Definition 1

— Let A be an algebra over K; then A is said to be diagonalizable if there exists an integer $n \geq 0$ such that A is isomorphic to the product algebra $K^n$. We say that A is diagonalized by an extension L of K if the algebra $A_{(L)}$ over L derived from A by extension of scalars is diagonalizable. We shall say that A is etale if there exists an extension of K which diagonalizes A.

We recall that the product algebra $K''$ is the vector space $K''$ equipped with the product defined by

If $\varepsilon_1, \ldots, \varepsilon_n$ is the canonical basis of $K^n$, we have

and $1 = \varepsilon_1 + \ldots + \varepsilon_n$.

Every etale algebra over $\mathbf{K}$ is commutative and of finite degree over $K$.