Proposition 1

— Let $A$ be a commutative algebra of finite degree over $K$. Then the following conditions are equivalent: a) The algebra $A$ is etale. b) There exists a basis of $A$ whose discriminant is non-zero. c) For each $x \neq 0$ in $A$ there exists $y$ in $A$ such that $\text{Tr}_{A/K}(xy) \neq 0$. Further, when these conditions are satisfied, the discriminant of any basis of $A$ is non-zero.

We shall show that when $A$ is assumed etale, the discriminant of $A$ with respect to any basis $(x_1, \ldots, x_n)$ of $A$ over $K$ is non-zero; this will in particular establish the implication $a) \Rightarrow b)$. By (4), with the above notation, it suffices to show that the matrix U is invertible, or equivalently, that the system of linear equations

has only the solution $\lambda_1 = \cdots = A, = 0$ in L. Now the relation (5) implies $\sum_{i=1}^n \lambda_i u_i(x) = 0$ for all $x \in A$, whence $\lambda_i = 0$ for $1 \leq i \leq n$, by the theorem on the linear independence of homomorphisms (V, p. 27, Th. 1).

The equivalence of b) and c) is a consequence of the following general lemma: