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Common Golden Cubic Fields and Their Discriminants

Abstract

The common fields generated by positive cube roots of golden blocks have explicit discriminants and Galois action.

Theorem 1.1 (Signatures, discriminants and the inversion action).

Lean statement: D5/S3/Factorization/Galois/GoldenCubicBlockCommonDiscriminants.golden_cubic_block_common_field_discriminants

Proof. Machine-checked in Lean as D5/S3/Factorization/Galois/GoldenCubicBlockCommonDiscriminants.golden_cubic_block_common_field_discriminants (✓ std3). ∎

Source. Repository-derived.

Commentary.

Write B_j for the golden Lucas number at index 3^j squared plus three. For a positive integer J, let F be the rational field generated by the positive cube roots of the first J golden blocks, and let N also contain a primitive cube root of unity. For each block, divide by c cubed, where c is the largest positive integer whose cube divides that block, and take the product of its distinct prime divisors. Let R be the product of these radicals over the first J blocks.

The rational degrees of F and N are 3^J and 2 times 3^J. F has one real place and (3^J - 1)/2 complex places. Every element of its designated embedding is real, while F is not totally real. Complex conjugation on N fixes a field rationally isomorphic to F, and N has degree two over that fixed field.

The absolute discriminant of N is 3^(3^J) times R^(4 times 3^(J-1)); that of F is 3^((3^J-1)/2) times R^(2 times 3^(J-1)). The root discriminant of N is sqrt(3) times R^(2/3). These are formulas for the same actual fields and the same radical product.

N is Galois over the rationals. Its Galois group is the semidirect product of J copies of the cyclic group of order three with the cyclic group of order two. The nontrivial order-two element acts by inversion on every cubic coordinate.

The local calculation at three identifies each completion with the cubic cyclotomic base completion, giving ramification index two and residue degree one. At a prime in R, the inertia group has order three; all other primes away from three are unramified. The different ideal identity and its absolute norm give the discriminant of N. Conjugation fixes exactly one prime above three, so the relative different over F has absolute norm three. The tower discriminant formula then gives the discriminant of F.

References