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Main discovery Two cyclic number fields have isometric integral trace lattices exactly when they share three arithmetic invariants: degree, real signature, and signed discriminant. The integral trace lattice means the ring of integers O_K equipped with the bilinear form (x,y) ↦ Tr_{K/ℚ}(xy).
How the result is proved The proof separates arithmetic reconstruction from lattice comparison. At odd primes p, the field degree together with the p-adic discriminant exponent v_p(|Disc|) uniquely recovers the local cyclic character subgroup Y_{E,p}. This recovers local ramification data from global degree and discriminant exponents. At the dyadic prime p = 2, the same data determine the local subgroup only up to an even–odd (parity) ambiguity. The global field signature (the counts of real and complex places) resolves that ambiguity for cyclic fields, so Y_{K,2} is determined.
On the algebraic side, the argument uses Leopoldt’s description of the ring of integers. Leopoldt gives an associated order A_E in the group algebra over which O_E is free of rank one. This replaces the usual Z[G] normal-basis argument when wild ramification occurs.
The trace pairing itself is written explicitly by means of normalized character resolvents and Gauss sums. Concretely, formula (22) expresses Tr((aΘ_E)(bΘ_E)) as |G| times a sum over characters. The summand involves the Fourier transforms ̂a(χ) and ̂b(χ^{-1}), the factor χ(−1), the conductor f_χ, and the intersection degree d_{E,m}. See equation (22) for the full expression.
A wider character-group statement More generally, for finite abelian extensions E and F, any isomorphism between their Dirichlet character groups that preserves every character’s conductor and parity induces an isometry between the integral trace lattices. Moreover this isometry can be chosen equivariant with respect to the dual identification of Galois groups. In the cyclic case, degree, signature and discriminant determine the character group together with conductor and parity, giving the full classification.
Concrete consequences The theorem proves a conjecture of Bolaños for wildly ramified cyclic fields. It extends the previously known tame classification by showing that, in general, signature must be retained as an additional invariant. In the tame (unramified at 2) case, the degree and discriminant already determine signature, so degree plus discriminant suffice.
Supported limitations and examples The signature hypothesis cannot be dropped. There are cyclic quartic fields K_+ = ℚ(√2+√2) and K_− = ℚ(√−2−√2) with the same signed discriminant 2^11 but different signatures: K_+ is totally real and K_− is totally imaginary. Their trace forms have different real signatures and are not isometric.
The parity-preservation requirement is also necessary. An isomorphism of character groups that preserves conductors but flips parity need not induce an isometry of integral trace lattices. The quartic example exhibits this failure.
Scope of the cyclic hypothesis The assumption that fields are cyclic is used only to deduce, from degree, discriminant and signature, a conductor‑and‑parity preserving identification of their character groups. The comparison theorem that a conductor‑and‑parity preserving character-group isomorphism gives a trace-lattice isometry does not require the Galois group to be cyclic.
Summary For cyclic number fields, the integral trace lattice is fully determined by degree, signature and signed discriminant. The proof recovers local character data from discriminant exponents, resolves a dyadic parity ambiguity using signature, and then compares lattices via explicit character-resolvent formulas and Leopoldt’s associated order. The result holds in both tame and wild settings, but the signature and parity conditions are essential and cannot be omitted.