arXiv:2608.11864v2

Equalities among simplest quartic fields: a complete classification

Zhi-Lin Zhang

math.NT11R1611J7011J68

Paper metadata

Original submission: August 12, 2026
Latest revision: August 15, 2026 · version 2
Claimed MSC: 11R16, 11J70, 11J68
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 8 pages; added references to earlier work of M.-N. Gras and clarified historical attribution; added a declaration of AI use; main results unchanged

Abstract

For a positive integer $n$ , let $f_n(X)=X^4-nX^3-6X^2+nX+1$ and let $K_n=\mathbb{Q}(ρ_n)$ , where $ρ_n$ is a root of $f_n$ . We determine all coincidences among these fields: for distinct positive integers $m,n$ , $K_m=K_n \Longleftrightarrow \{m,n\}\in\{\{1,103\},\{2,22\},\{4,956\}\}$ . Thus the three previously known equalities are the only ones. This extends Hoshi's finite-range classification to all positive integral parameters and, in particular, subsumes the uniqueness results of Pincus and Washington. The proof combines Hoshi's correspondence between equal simplest quartic fields and primitive solutions of a quartic Thue equation with estimates of Lettl--Pethő--Voutier for rational approximations to two of its real roots. A Gaussian-integer identity yields a lower bound for the denominator of the resulting rational approximation; the continued-fraction information and the approximation estimates then exclude every parameter exceeding $1000$ .

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 19, 2026

01Statements2 reported findingsCorrect

The complete classification of equal simplest quartic fields and its two stated corollaries are correct under the cited Gras, Hoshi, and Lettl–Pethő–Voutier inputs.

Theorem 1.1Correct

Complete classification of equal simplest quartic fields

Pages 2 and 7 · Theorem 1.1 and Section 6 · arXiv:2608.11864v2

For distinct positive parameters, put $a=\min\{m,n\}$ and $b=\max\{m,n\}$ . The cited Gras–Hoshi irreducibility and reducibility results separate the quadratic case $a=3$ or $b=3$ . If $a\leq1000$ , Hoshi's complete list of primitive solutions gives exactly $(a,b)=(1,103),(2,22),(4,956)$ . If $a>1000$ , Proposition 5.1 excludes $K_a=K_b$ . Hoshi's recorded identities supply the converse for the three listed pairs, so both implications and the full positive-integral parameter range are established.

Hoshi, field-isomorphism correspondence and finite-range tables ↗

Corollaries 6.1–6.2Correct

Uniqueness and the odd-parameter specialization

Page 7 · Corollaries 6.1–6.2 · arXiv:2608.11864v2

The three unordered exceptional pairs in Theorem 1.1 are disjoint, which gives at most one distinct positive partner for every parameter. Among those pairs, only $\{1,103\}$ contains odd parameters, yielding the stated odd-parameter equivalence immediately. No additional hypothesis is introduced in either deduction.

02Proofs3 reported findingsCorrect

The proof chain is correct and complete: Hoshi's correspondence is applied in its primitive odd-divisor form, the Gaussian-integer identity is exact, and the continued-fraction estimates exclude every smaller parameter above $1000$ .

Proposition 3.1Correct and complete

Thue-equation reduction and Gaussian-integer identity

Pages 4–5 · Proposition 3.1, Equations (16)–(18) · arXiv:2608.11864v2

For $a,b\neq3$ , the root fields are the cyclic quartic splitting fields to which Hoshi's criterion applies. Its parity refinement supplies coprime $x,y$ of opposite parity and an odd divisor $c\mid a^2+16$ with $F_a(x,y)=c$ and $N=\pm b$ . The symmetries of $F_a$ give $y>0$ and $0<|x|<y$ without changing $P=xy(x^2-y^2)$ or $N$ . With $R=x^4-6x^2y^2+y^4$ , the identities $c=R-aP$ and $N=(aR+16P)/c$ give $(4+ia)(x+iy)^4=c(4+iN).$ Taking norms yields exactly $(a^2+16)(x^2+y^2)^4=c^2(b^2+16)$ .

Hoshi, Theorem 1.4 and Lemma 6.1 ↗

Proposition 5.1Correct and complete

Exclusion of equal fields for $1000<a<b$

Pages 5–7 · Proposition 5.1 · arXiv:2608.11864v2

The norm identity gives $|c|<(x^2+y^2)^2$ and Hoshi's divisor condition gives $|c|\leq a^2+16$ . Separating the signs of $x$ in $c=R-aP$ proves $y>a/4$ , hence the denominator threshold in the specialized Lettl–Pethő–Voutier theorem. The explicitly listed early convergents to the two relevant roots all have $|F_a(x,y)|>a^2+16$ . Every later convergent to the root near $1$ has $y\geq a^2/11$ , contradicting $y^{5/4}<(10/9)(a^2+16)$ ; every later convergent to the root near $-1/a$ has $y\geq a^2/6$ , contradicting $y^{5/4}<a^2+16$ . The numerical inequalities used in both contradictions hold throughout $a>1000$ , and the even and odd continued-fraction cases are both covered.

Lettl–Pethő–Voutier, Theorem 2 and Lemma 9 ↗

Proposition 4.1 and Section 6Correct and complete

Finite range and final case split

Pages 4 and 7 · Proposition 4.1 and proof of Theorem 1.1 · arXiv:2608.11864v2

Hoshi's Theorem 8.1 and Tables 2–3 classify every nontrivial primitive solution for a fixed parameter at most $1000$ ; retaining the positive second parameter larger than the first gives precisely the three ordered pairs used here. The final proof then exhausts $a=3$ , $a\leq1000$ , and $a>1000$ , and invokes Hoshi's explicit equalities for the reverse implication.

Hoshi, Theorem 8.1 and Tables 2–3 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.11864v2
Authors listed: Zhi-Lin Zhang
Audit date: August 19, 2026

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Equalities among simplest quartic fields: a complete classification

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Complete classification of equal simplest quartic fields

Uniqueness and the odd-parameter specialization

Thue-equation reduction and Gaussian-integer identity

Exclusion of equal fields for 1000<a<b1000<a<b1000<a<b

Finite range and final case split

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Exclusion of equal fields for $1000<a<b$