# Étale algebra

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2010 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
How to Cite This Entry:
Étale algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=%C3%89tale_algebra&oldid=51468