Scalar Quotients of Oriented Eisenstein Factors
Abstract
An odd-parameter Eisenstein factor has a scalar contraction equal to its norm ideal and a scalar quotient isomorphism.
Theorem 1.1 (The oriented factor contracts to its norm ideal).
Proof. Machine-checked in Lean as D5/S3/Factorization/QuadraticIdeals/EisensteinOddQuotient.eisenstein_odd_scalar_quotient (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let E be the quadratic algebra Z[omega] with omega squared plus omega plus one equal to zero. For an odd natural number b, set eta equal to -2 + b omega and B equal to b squared plus 2b plus 4.
If eta times (u + v omega) is scalar, its omega coefficient gives bu = (b + 2)v. Oddness makes b and b + 2 coprime, so u = (b + 2)t and v = bt. The scalar product is then -Bt. Conversely, an explicit multiple of eta has scalar value B, giving the exact contraction.
The scalar map to E modulo eta has kernel BZ. Also b and B are coprime: B is congruent to four modulo b and b is odd. In the quotient, b omega = 2 and B = 0. Bezout coefficients therefore express omega as a scalar class. Every quotient class has a representative u + v omega, so the scalar map is surjective and induces the displayed ring equivalence with the stated action on integers.