View source for Étale algebra

{{TEX|done}}{{MSC|13B|12F}}

A commutative algebra $A$ finite-dimensional over a field $K$ for which the bilinear form induced by the trace
$$
\langle x,y \rangle = \mathrm{tr}_{A/K} (x\cdot y)
$$
is non-singular.  Equivalently, an algebra which is isomorphic to a product of fields $A \sim K_1 \times \cdots \times K_r$ with each $K_i$ a [[field extension]] of $K$.

Since $\langle xy,z \rangle = \mathrm{tr}(xyz) = \langle x,yz \rangle$, an étale algebra is a [[Frobenius algebra]] over $K$.

====References====
* Tsit-Yuen Lam, "Lectures on Modules and Rings", Graduate Texts in Mathematics '''189''', Springer (2012) {{ISBN|1461205255}} {{ZBL|0911.16001}}