Mixed-Radicand Prime Saturation
Abstract
A mixed pure-cubic lattice is saturated at primes simple in either cubic radicand.
Theorem 1.1 (The two local cubic generators).
Lean statement: D5/S3/Arith/Lattices/PureCubicMixedPrimeSaturation.pure_cubic_mixed_prime_saturation
Proof. Machine-checked in Lean as D5/S3/Arith/Lattices/PureCubicMixedPrimeSaturation.pure_cubic_mixed_prime_saturation (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let alpha generate a cubic field with alpha cubed equal to m times n squared. Put beta = alpha squared divided by n, and gamma = (1 + c alpha + v beta)/3, where v is 1 or -1. The integers also satisfy c cubed times m n squared = 1 + 9a and c squared times n = v + 3k. The lattice S is the integer span of 1, alpha, and gamma.
For a prime p occurring exactly once in m n squared and coprime to 3n, or exactly once in m squared n and coprime to 3m, an integral element z with p z in S already belongs to S.
In the first case the minimal polynomial of alpha is Eisenstein at p, and 3n times S lies in the ring generated by alpha. In the second case beta cubed is m squared n, beta squared is m alpha, and 3m times S lies in the ring generated by beta. Each Eisenstein criterion returns the cleared element to its generated ring; cubic reduction returns that ring to S. Coprimality of p with the clearing factor then removes the factor by Bezout’s identity.