Étale algebra

2020 Mathematics Subject Classification: Primary: 13B Secondary: 12F [MSN][ZBL]

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