Complete Golden Cubic Cyclotomic Intersection
Abstract
The complete golden cubic radical field and its actual cyclotomic field meet only in their cubic cyclotomic base.
Theorem 1.1 (The complete radical field has trivial intersection over E).
Lean statement: D5/S3/Factorization/Galois/GoldenCubicCompleteCyclotomicDisjointness.actual_complete_cubic_cyclotomic_disjointness
Proof. Machine-checked in Lean as D5/S3/Factorization/Galois/GoldenCubicCompleteCyclotomicDisjointness.actual_complete_cubic_cyclotomic_disjointness (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every natural number j, let E be the third cyclotomic field over the rationals, L an algebraic closure of E, and m = 80 * 3^(j + 2). A primitive m-th root zeta in L generates the actual cyclotomic field C over E. The support S contains the prime factors of blockNorm(toNat(goldenLucas(3^i) - 1)) for each index 1 <= i < j.
The same primary Eisenstein factor pi_p is used for every occurrence of p in S. Its principal ideal is prime, its norm is p, it is one modulo three, and p is one modulo three. It is coprime to its conjugate; factors attached to different primes are pairwise coprime in all four primary and conjugate combinations. These factors give the oriented product of each earlier block with the exponent equal to the p-adic valuation of the Fibonacci number at the entry rank of p.
Choose the cubic roots of each primary factor and its conjugate, and of two and three, in that same closure L. Their cubes are exactly the corresponding selected radicands. The field M generated by this complete root family has degree three to the power two times the size of S plus two over E. It is Galois over E, and no element of M cubes to the distinguished primitive third root of unity. Its automorphism group is equivalent to one copy of the cyclic group of order three for each root, with the stated coordinatewise multiplication action on those roots.
For these very same roots and the chosen primitive m-th root, the intersection of M with C is E. The m-th cyclotomic group over E has a unique cubic quotient. Its corresponding nontrivial degree-three subfield is generated by a primitive ninth root of unity, and contains a cube root of the distinguished cubic root of unity. The latter cannot lie in M, so no nontrivial common Galois subfield remains.