The Integral Third-Cyclotomic Model of Eisenstein Integers
Abstract
The Eisenstein quadratic order and the integral third cyclotomic ring are equivalent as integer algebras.
Theorem 1.1 (The two integral algebras are equivalent).
Proof. Machine-checked in Lean as D5/S3/Factorization/QuadraticIdeals/EisensteinCyclotomicBridge.eisenstein_cyclotomic_equiv_exists (✓ std3). ∎
Source. Repository-derived.
Commentary.
E is the quadratic integer algebra generated by omega with omega squared plus omega plus one equal to zero. The target is the ring of integers of the third cyclotomic field.
The basis 1, omega makes E a degree-two power-basis algebra. A primitive third root gives an integral power basis of the cyclotomic ring, also of degree two. Each generator has minimal polynomial X squared plus X plus one, so the power-basis equivalence identifies the algebras.