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The paper aims to resolve the form-class question for non-cyclic totally real cubic number fields using trace-zero lattice invariants.

February 25, 2026Open Access

Form-Class Resolution for Non-Cyclic Totally Real Cubics: The Trace-Zero Lattice Invariant (Paper 66 of the Constructive Reverse Mathematics Series)

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The paper aims to resolve the form-class question for non-cyclic totally real cubic number fields using trace-zero lattice invariants.
Constructs a trace-zero sublattice for totally real cubic fields.
Proves a determinant identity involving the Gram matrix of this lattice.
Enumerates non-cyclic totally real cubics and validates their properties up to a certain discriminant.
Applies explicit computational methods to affirm results.
Establishes that the determinant of the Gram matrix equals 3 times the discriminant of the field.
Finds all reduced trace-zero forms among non-cyclic cubics are distinct.
Demonstrates that no simpler predictor exists for the trace-zero form's properties.

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Paper examines trace-zero lattice invariants in non-cyclic totally real cubics, revealing implications for number theory.

Paul Chun-Kit Lee (2026) studied this question.

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