Golden Cubic Compatibility Conjugacy
Abstract
The actual compatibility automorphism has a two-element conjugacy class over the rationals.
Theorem 1.1 (Two rational conjugates and the Galois-group order).
Lean statement: D5/S3/Factorization/Galois/GoldenCubicCompatibilityConjugacy.actual_conjugacy_data
Proof. Machine-checked in Lean as D5/S3/Factorization/Galois/GoldenCubicCompatibilityConjugacy.actual_conjugacy_data (✓ std3). ∎
Source. Repository-derived.
Commentary.
For the actual radical and cyclotomic compositum, complex conjugation changes the selected automorphism on the real cube root of two. Automorphisms fixing the cubic cyclotomic base form an abelian subgroup of index two. Thus the rational conjugacy class consists of the selected automorphism and its distinct conjugate.
The class has size two. Linear disjointness of the actual radical and cyclotomic fields gives rational Galois-group order equal to the totient of the modulus times 3 to the power twice the earlier support size plus two.