Quadratic étale algebras

A quadratic algebra QuadraticAlgebra R a b over a commutative ring is étale if and only if its discriminant b ^ 2 + 4 * a is a unit, including in characteristic two. This is the finite-free discriminant criterion for the basis 1, ω. Over a field the condition is that the discriminant is nonzero.

References

• Stacks Project, Lemma 49.3.1.

theorem QuadraticAlgebra.etale_iff_isUnit_discr {R : Type u_1} [CommRing R] {a b : R} : Algebra.Etale R (QuadraticAlgebra R a b) ↔ IsUnit (discr a b)

A quadratic algebra over a commutative ring is étale if and only if its discriminant is a unit.

theorem QuadraticAlgebra.etale_of_discr_ne_zero {K : Type u_1} [Field K] {a b : K} (hd : discr a b ≠ 0) : Algebra.Etale K (QuadraticAlgebra K a b)

A quadratic algebra over a field with nonzero discriminant is étale.