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First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals

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Pith's one-line read First-degree prime ideals of Z[alpha+beta] can be combined from those of Z[alpha] and Z[beta], but the claimed preservation of divisibility of principal ideals rests on a false intersection formula.

arxiv 1908.00383 v3 pith:MBUH7HNN submitted 2019-08-01 math.NT

Giordano Santilli , Daniele Taufer This is my paper · ORCID
classification math.NT
keywords idealsprimefirst-degreebiquadraticcorrespondencefieldsprincipalapplications
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

Number fields have rings of integers, and inside them sit smaller rings generated by one element, called orders. This paper studies biquadratic fields, made from two quadratic fields, and their order Z[gamma] where gamma is the sum of the two quadratic generators alpha and beta. A first-degree prime ideal in such an order is an ideal of prime norm, and can be identified with a root of the defining polynomial modulo that prime. The paper shows that such primes in the biquadratic order are almost exactly sums of matching primes in the two quadratic orders. This part is correct and is a clean structural observation. The paper then tries to show that if such primes divide a special principal ideal in the quadratic orders, their sum divides the corresponding principal ideal in the biquadratic order. This requires knowing exactly which elements of the quadratic order lie in the biquadratic ideal. The paper's Proposition 4.1 claims a simple formula for that intersection, but the formula is false. In the paper's own Example 2, the claimed generator 15+10alpha is not even in the biquadratic order Z[gamma], so it cannot lie in the ideal. The unique quotient would be 5+2i-sqrt(6), which is not an integer polynomial in gamma. Thus the intersection formula fails, and the divisibility theorems built on it are not proven.
Extended reading notes

Core claim

The paper's central advertised claim is Theorem 4.2: if (r,p) is a first-degree prime ideal of Z[alpha] dividing I intersect Z[alpha] and (s,p) is one of Z[beta] dividing I intersect Z[beta], then the combination (r+s,p) divides I = <n+m*gamma>, except when p != 2, n = 0 mod p, and r+s != 0 mod p. The converse is Theorem 4.3. These theorems rest on Proposition 4.1, which asserts I intersect Z[alpha] is the principal ideal generated by g = (n+m*alpha+m*beta)(n+m*alpha-m*beta).

Load-bearing premise

The load-bearing unstated assumption is that the factor n+m*alpha-m*beta belongs to Z[gamma], which would make g = (n+m*gamma)(n+m*alpha-m*beta) an element of the ideal I. In general Z[gamma] is a proper sublattice of the Z-span of {1, alpha, beta, alpha*beta}, and alpha and beta are not in Z[gamma]. The assumption is false for the paper's own Example 2, where a=-4, b=6, n=5, m=1, since g=15+10*alpha is not in Z[gamma] and hence not in I. This premise enters in Proposition 4.1 and is used throughout Section 4.

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Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It relies on standard algebraic number theory and an implicit, false membership assumption that undermines Section 4. The Section 3 results stand on the first two axioms.

assumptions (3)
  • standard math Standard characterization of first-degree prime ideals via evaluation maps (Theorem 2.2, after Buhler-Lenstra-Pomerance).
    Used throughout to identify first-degree prime ideals with roots of the defining polynomial modulo p.
  • domain assumption All work is done in the order Z[gamma] generated by gamma, not the full ring of integers of the biquadratic field.
    Definition 2.1 and Theorem 2.2 apply to Z[theta]. This limits the results to orders, which is stated.
  • ad hoc to paper The element n+m*alpha-m*beta belongs to Z[gamma], so that g=(n+m*gamma)(n+m*alpha-m*beta) lies in the ideal I.
    This unstated assumption is needed in Proposition 4.1 to show the generator is an element of I. It is false in general; in Example 2, n+m*alpha-m*beta is 5+2i-sqrt(6), which is not in Z[gamma].

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Pith. "Pith review of First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals." pith.science (2026). https://pith.science/paper/MBUH7HNN

@misc{pith2026190800383,
  author       = {Pith},
  title        = {Pith review of: First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBUH7HNN}},
  note         = {Machine review of arXiv:1908.00383}
}
read the original abstract

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.

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Works this paper leans on

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