Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Counting biquadratic number fields with quaternionic and dihedral extensions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves asymptotic formulas for the number of biquadratic number fields of bounded discriminant that embed into quaternionic or dihedral extensions, and shows that the density of such fields is zero.

desk verdict Solid, detailed counting argument for a natural arithmetic-statistics question; the main theorem is plausible and the architecture is sound, but the proof leans on a delicate Siegel-Walfisz-type lemma and a copy-paste error in Lemma 4's even case needs fixing. read the letter →

arxiv 2506.21522 v1 pith:FV2T54CR submitted 2025-06-26 math.NT

Louis M. Gaudet , Siman Wong This is my paper · ORCID
classification math.NT MSC 11R3211R4511N37
keywords biquadraticfieldsquaterniongroupQ8dihedralD4inverseGaloisproblemHilbertsymbolslinkedquadraticcharactersdiscriminantasymptoticsextension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to count, among biquadratic number fields of discriminant at most X, how many can be placed inside a Galois extension with Galois group the quaternion group Q8 or the dihedral group D4. The main theorem gives an asymptotic formula for each group and each real or complex signature, with explicit leading constants, and shows that the density of such fields is zero in every case. This matters because Q8 and D4 are the two smallest groups whose extension problem over Q is genuinely obstructed, so the count measures how often a prescribed intermediate field can be realized inside a solution to the inverse Galois problem. The proof converts classical Hilbert-symbol criteria for embeddability into sums of linked quadratic characters, then evaluates those sums with existing analytic machinery.

What carries the argument

The argument is carried by three reductions. First, biquadratic fields are parameterized by admissible triples of squarefree integers, with the discriminant expressed through the three quadratic subfields and a type index modulo 4. Second, the classical criteria for embeddability in Q8 and D4 are rewritten as local Hilbert-symbol conditions, which expand into products of Kronecker symbols; a divisor-sum transformation turns these into weighted sums over six pairwise coprime squarefree variables D_ij of the linked character product g(D) = (D22D32/D11)(D12D32/D21)(D12D22/D31). Third, a single averaging lemma evaluates the resulting sums G_w(x; Psi) with main term proportional to $\sqrt$(x)(log $\sqrt$(x))^{1/2}, using dyadic decomposition, a bilinear-form bound for Jacobi symbols, a uniform estimate for weighted quadratic character sums with modulus up to a power of log x, and a Dirichlet-series method. The final constants come from finite sums over residue classes in (Z/8Z)^6.

What would settle it

Exhibit one totally imaginary biquadratic field that admits a Q8-extension; Lemma 1 and the theorem's c_-(Q8) = 0 predict that no such field exists, so a single example would disprove the classification and the count.

Watch

Extended reading notes

Core claim

The paper establishes that for $\sigma$ in {+,-} and G = Q8 or D4, the number B_sigma(X; G) of biquadratic fields of discriminant at most X and signature $\sigma$ admitting a G-extension satisfies $$B_\$\sigma$(X;G) = \frac{c_\$\sigma$(G)}{\sqrt{2\pi}}\sqrt{X}\,(\log X)^{1/2}\prod_p\left(1-\frac1p\right)^{3/2}\left(1+\frac{3}{2p}\right) + O\bigl(\sqrt{X}\,(\log X)^{1/4}\bigr),$$ with c_+(Q8)=25/168, c_-(Q8)=0, c_+(D4)=33/56, and c_-(D4)=33/28. In particular, the proportion of biquadratic fields admitting such extensions tends to zero for both groups. The zero value for totally imaginary Q8 fields reflects a structural fact already visible in the local criteria: no totally imaginary biquadratic field satisfies the Hilbert-symbol conditions for a Q8-extension.

Load-bearing premise

The central assumption is a uniform estimate for certain weighted sums of quadratic characters over squarefree integers, valid for moduli up to a power of log x but with an ineffective constant; the proof needs this estimate to hold at the chosen cutoff parameter z = exp((log x)^{1/12}), and if it fails there the main term collapses.

Editorial extensions

If this is right

  • For both Q8 and D4, the number of biquadratic fields of discriminant at most X admitting the relevant extension is asymptotic to a constant times sqrt(X)(log X)^{1/2}, so the density among all biquadratic fields tends to zero.
  • No totally imaginary biquadratic field admits a Q8-extension, so B_-(X; Q8) = 0 for every X.
  • The nonzero constants are ordered c_-(D4) = 33/28 > c_+(D4) = 33/56 > c_+(Q8) = 25/168, so both the signature and the choice of group affect the density.
  • The same character-sum machinery yields an independent proof that the full set of biquadratic fields has size asymptotic to a constant times sqrt(X)(log X)^2, matching the known count.
  • The error term O(sqrt(X)(log X)^{1/4}) is smaller than the main term by a factor of (log X)^{-1/4}, leaving the explicit leading constants as the main numerical content.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same reduction should give a C4 x C2 analogue of Theorem 1, since the cyclic case reduces to a single norm condition rather than a product of Hilbert symbols.
  • A testable extension is to interpret the residue-class sums producing constants such as 25/168 and 33/28 as a local probabilistic model for the probability that a random biquadratic field admits each extension; that model would be a finite computation independent of the analytic lemma.
  • Because the uniform character-sum estimate has an ineffective constant, the theorem cannot currently yield explicit ranges of X where the main term dominates; removing that ineffectivity is a natural next step.
  • The structural vanishing at c_-(Q8) suggests a family-level pattern: for 2-groups whose extension criteria force both discriminants to be positive, the complex-signature count may vanish at this level of the extension lattice.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper establishes asymptotic formulae for the number of biquadratic number fields of bounded discriminant that admit a Q8-extension or a D4-extension, with explicit leading constants. The authors parametrize biquadratic fields by admissible triples of squarefree integers, translate Witt's and the classical norm criteria into local Hilbert-symbol and Kronecker-symbol conditions, convert the counting functions into sums of linked quadratic characters, and then adapt Heath-Brown's method to evaluate the resulting character sums. The main theorem states that B_sigma(X;G) is of order sqrt(X)(log X)^{1/2} with explicit constants c_sigma(G), thereby showing that the density of embeddable biquadratic fields is zero. The paper also contains a new proof of the earlier asymptotic count of all biquadratic fields and a discussion connecting the problem to projective Artin representations.

Significance. If the main theorem is correct, it provides the first discriminant-ordered asymptotic formula for these inverse-Galois extension problems over Q, complements the parameter-counting results of Fouvry-Luca-Pappalardi-Shparlinski, and parallels Rome's result on the Hasse norm principle with the same Euler product. The paper is careful to identify the exact finite sums over residue classes modulo 8 that determine the constants, and it explicitly credits the correction of a known parameter error in Heath-Brown's method. The structure of the proof is clear and the arithmetic reinterpretation in the appendix is valuable. The main obstacle is the correctness of the analytic lemmas, in particular Lemma 11, whose statement is false as written and which is used in the proof of Proposition 3.

major comments (3)
  1. [Section 9, Lemma 11, Eq. (31)] Lemma 11 is false as stated. The asymptotic (31) applies to every triple of characters modulo 8, but the pole order of the associated Dirichlet series F(s) depends on how many of the chi_i are of polar type (chi_0 for delta_i=0, and chi_0 or chi_{-1} for delta_i=1), not merely on the sum of the delta_i. For example, take delta_1=delta_2=delta_3=0, chi_1=chi_2=chi_0, and chi_3 a nonprincipal character modulo 8. Then F(s)=prod_p(1+(1+chi_3(p)/2)p^{-s}) has a simple pole at s=1, so the summatory function in (31) is of size x, whereas (31) with Upsilon=0 and J=1/2 gives O(x(log x)^{-1/2}). This is not a harmless overestimate: the error term is smaller than the actual main term. Since Proposition 3 applies Lemma 11 to every triple of characters modulo 8 in the evaluation of G1_w and G2_w, the proof of the main term of Proposition 3, and hence of the constants in Theorem 1, is not valid as written.
  2. [Section 4.2, Lemma 4, even case] In the case 2|gcd(d1,d2), the displayed formulas for 1_{M1}1_{M2} and 1_{M1}1_{M3} are identical: both have the factor psi(D2D3) and the same sign condition sigma != (pm1,-1). The odd-case derivation and the final formula for Feven(D;D4) show that the first identity should instead involve psi(D1D2) with the condition sigma != (pm1,mp1). The final statement of Lemma 4 appears to be the correct one, and Proposition 2 is consistent with the corrected version, so the error is localized to the proof, but the proof as printed is internally inconsistent and must be repaired.
  3. [Section 9, Lemma 9 and Section 6, Lemma 6] Lemma 9 is the load-bearing analytic estimate that produces the sqrt(X)(log X)^{1/2} order, yet its extension to the case delta=1 is asserted in a single sentence as 'completely analogous' to Heath-Brown's Lemma 6. This is not adequate at the level of uniformity used here. In the delta=1 case the relevant Dirichlet series involves L(s,chi)L(s,chi chi_{-1}) under a square root, and one must control the zero-free region and possible exceptional zeros for both L-functions at moduli as large as (log x)^N with N=11520. The footnote on p.24 correctly records that Heath-Brown's original parameter choice was erroneous, so this point is genuinely delicate. Please provide a complete proof of Lemma 9, or a precise reference that covers the delta=1 statement with the stated uniformity and ineffective constant.
minor comments (5)
  1. [Section 6, proof of Lemma 5] In the displayed inequality after applying Lemma 8, the factor Aij Akell appears to be counted twice: after the outer sum over the remaining variables the total should be (prod Auv) {min(Aij,Akell)}^{-1/32}, which gives the stated final bound sqrt(x) {min(...)}^{-1/32}. The extra displayed factor should be corrected.
  2. [Section 6, proof of Proposition 3] The definitions of G1_w and G2_w appear to be interchanged relative to the preceding paragraph: after noting that A11,A21,A31 >= z forces D12=D22=D32=1, the recombined sum should select the second column equal to 1, not the first. Since the final Y(w;Psi) is symmetric in the two columns, the result is unaffected, but the exposition should be aligned.
  3. [Section 7, Theorem 4] The finite sums S_sigma_h(G) are reported as computer computations without an accompanying verification. Since these values determine the final constants, please provide the code, a table of intermediate values, or another reproducible check.
  4. [Section 8, proof of Theorem 5] The sentence 'This asymptotic gives the result, after summing over the relevant sigma' is confusing because sigma denotes a pair of signs in the proof and a field sign in the theorem. It would be clearer to state separately the totals for the totally real case and for the three totally complex cases.
  5. [Appendix, A.3] There is a typo: 'quanternionic' should be 'quaternionic'. There are a few similar typographical slips elsewhere (for example 'arrangements' is misspelled in Section 1.4), which should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main constants arise from Hilbert-symbol criteria and Landau-Selberg-Delange analysis, not from fitting or self-referential definitions.

full rationale

The derivation chain is self-contained and the claimed asymptotic formulae are not built from their own outputs. The solvability criteria in Lemmas 1 and 2 are classical algebraic facts (Witt's criterion and Hasse norm theorem), converted into explicit quadratic-symbol conditions; the characteristic functions in Lemmas 3 and 4 are derived, not postulated to match Theorem 1. The counting reduction in Propositions 1 and 2 expresses each B_sigma_h(X;G) as finite sums over residue classes modulo 8 of explicit functions f and r times a common character sum G_w(x; Psi). Proposition 3 evaluates this sum via Heath-Brown's linked-character method, and its main term is produced by the Landau-Selberg-Delange constant kappa_0 in Lemma 11, with the finite sums S_sigma_h(G) computed independently over H^6. None of these ingredients is fitted to the target counts; the proof-internal cutoffs z, theta, and N are auxiliary and cancel from the final statement. The only delicate point, Lemma 9's Siegel-Walfisz-type uniformity at q <= (log x)^N with ineffective constant, is an imported external analytic estimate whose possible failure would be a correctness risk, not circularity: the authors explicitly flag the parameter-choice issue via Fouvry-Klueners and Xiong-Zaharescu rather than appealing to their own prior results. There are no self-citations used as load-bearing evidence, no renamed known results presented as new structure, and no fitted quantities relabeled as predictions. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The theorem is derived from standard local-global principles and analytic estimates; no constants are fitted to data. The two proof-internal cutoffs theta and N are listed for completeness but cancel from the final statement. No new physical or algebraic entities are postulated.

free parameters (2)
  • theta (dyadic cutoff exponent) = 1/12
    Chosen in Section 6 to balance error terms in Lemmas 6 and 7. It is an auxiliary proof parameter and does not appear in the final asymptotic.
  • N (uniformity exponent in Lemma 9) = 11520
    Determined by the requirement N*theta > 960. Used only to keep the modulus within the Siegel-Walfisz uniformity range; it cancels from the final result.
assumptions (5)
  • domain assumption Witt's criterion and the folklore D4 norm criterion characterize embeddability into Q8 and D4 extensions.
    Invoked as Theorems 2 and 3 in Section 3. These are classical results from quadratic form theory and inverse Galois theory, cited to Witt and to Jensen-Ledet-Yui.
  • standard math Hasse norm theorem and the local-global equivalence of quadratic forms via Hilbert symbols.
    Used throughout Section 3 to rewrite the local criteria as product formulas for Hilbert symbols over all places of Q.
  • standard math Lemma 9, a Siegel-Walfisz-type uniform estimate for sums of 2^(-omega(n)) chi(n) psi(n)^delta, with ineffective implied constant.
    This is the analytic engine of Lemma 6 and Proposition 3. The proof is said to be analogous to Heath-Brown's Lemma 6; the parameter z is chosen to avoid the known erroneous range.
  • standard math Heath-Brown's bilinear form estimate for the Jacobi symbol, Lemma 8.
    Used to eliminate subsums with two large linked variables. The statement is taken from Heath-Brown's congruent number paper.
  • standard math Landau-Selberg-Delange method, used in Lemmas 11 and 12.
    Provides the main terms for the character sums over squarefree integers and produces the constants kappa0 and lambda0.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Counting biquadratic number fields with quaternionic and dihedral extensions." pith.science (2026). https://pith.science/paper/FV2T54CR

@misc{pith2026250621522,
  author       = {Pith},
  title        = {Pith review of: Counting biquadratic number fields with quaternionic and dihedral extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FV2T54CR}},
  note         = {Machine review of arXiv:2506.21522}
}
read the original abstract

We establish asymptotic formulae for the number of biquadratic number fields of bounded discriminant that can be embedded into a quaternionic or a dihedral extension. To prove these results, we express the solvability of these inverse Galois problems in terms of Hilbert symbols, and then apply a method of Heath-Brown to bound sums of linked quadratic characters.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Counting $D_4$-field extensions by multi-invariants

    math.NT 2025-07 conditional novelty 6.0 of 10

    For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided ...

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Baily, On the density of discriminants of quartic fields

    A. Baily, On the density of discriminants of quartic fields. J. Reine Angew. Math. 315 (1980) 190-210

    work page 1980

  2. [2]

    Bhargava, The density of discriminants of quartic rings and fields

    M. Bhargava, The density of discriminants of quartic rings and fields. Ann. Math. 162 (2005) 1031-1063

    work page 2005

  3. [3]

    Bosma, J

    W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997) 235-265

    work page 1997

  4. [4]

    J. W. S. Cassels and A. Fr¨ ohlich,Algebraic number theory. Academic press, 1967

    work page 1967

  5. [5]

    Cohen, Number Theory, volume I: Tools and Diophantine Equations

    H. Cohen, Number Theory, volume I: Tools and Diophantine Equations . Springer-Verlag, 2007

    work page 2007

  6. [6]

    Cohen, F

    H. Cohen, F. Diaz y Diaz and M. Olivier, Counting discriminants of number fields. J. Th´ eorie des Nombres de Bordeaux 18 (2006) 573-593

    work page 2006

  7. [7]

    Feit, Characters of Finite Groups

    W. Feit, Characters of Finite Groups . Benjamin, 1967

    work page 1967

  8. [8]

    Fouvry and J

    ´E. Fouvry and J. Kl¨ uners, On the 4-rank of class groups of quadratic number fields. Invent. Math. 167 (3) (2007) 455-513

    work page 2007

Show all 37 references
  1. [9]

    Fouvry, F

    ´E. Fouvry, F. Luca, F. Pappalardi, and I. Shparlinski, Counting dihedral and quaternionic extensions. Trans. Amer. Math. Soc. 363 (2011) 3233-3253

    2011

  2. [10]

    Frederiksen and I

    P. Frederiksen and I. Kiming, Galois representations of dihedral type over Qp. Acta Arith. 111 (2004) 43-59

    2004

  3. [11]

    Friedlander and H

    J. Friedlander and H. Iwaniec, Ternary quadratic forms with rational zeros.J. Th´ eor. Nombres Bordeaux 22 (2010) 97-113

    2010

  4. [12]

    D. R. Heath-Brown, The size of Selmer groups for the congruent number problem. Invent. Math. 111 (1993) 171-195

    1993

  5. [13]

    C. U. Jensen, A. Ledet and N. Yui, Generic polynomials. Cambridge Univ. Press, 2002

    2002

  6. [14]

    C. U. Jensen and N. Yui, Polynomials with Dp Galois group. J. Number Theory 15 (1982) 347-375

    1982

  7. [15]

    Kiming, On the Liftings of 2-Dimensional Projective Galois Representations over Q

    I. Kiming, On the Liftings of 2-Dimensional Projective Galois Representations over Q. J. number theory 56 (1996) 12-35

    1996

  8. [16]

    Landau, ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate

    E. Landau, ¨Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. der Math. u. Phys. (3) 13 (1908), 305-312

    1908

  9. [17]

    https://groupprops.subwiki.org/wiki/Linear_ representation_theory_of_dicyclic_groups

    Linear representation theory of dicyclic groups. https://groupprops.subwiki.org/wiki/Linear_ representation_theory_of_dicyclic_groups

  10. [18]

    The LMFDB Collaboration, The L-functions and modular forms database https://www.lmfdb.org, 2025, [Online; accessed 3 April 2025]

    2025

  11. [19]

    M¨ aki, On the density of abelian number fields

    S. M¨ aki, On the density of abelian number fields. Thesis, Helsinki, 1985

    1985

  12. [20]

    Malle, On the distribution of Galois groups

    G. Malle, On the distribution of Galois groups. J. Number Theory 92 (2002) 315-329

    2002

  13. [21]

    Malle, On the distribution of Galois groups

    G. Malle, On the distribution of Galois groups. II. Experiment. Math. 13 (2004) 129-135

    2004

  14. [22]

    D. A. Marcus, Number fields, 2nd ed. Springer-Verlag, 2018

    2018

  15. [23]

    D. J. S. Robinson, A course in the theory of groups , 2nd ed. Springer-Verlag, 1996

    1996

  16. [24]

    D. E. Rohrlich, Artin representations of Q of dihedral type. Math. Res. Lett. 22 (2015) 1767-1789

    2015

  17. [25]

    D. E. Rohrlich, Quaternionic Artin representations of Q. Math. Proc. Cambridge Philos. Soc. 163 (2017) 95-114

    2017

  18. [26]

    D. E. Rohrlich, Quaternionic Artin representations and nontraditional arithmetic statistics. Trans. AMS 372 (2019) 8587-8603

    2019

  19. [27]

    D. E. Rohrlich, Dihedral Artin representations and CM fields. Proc. AMS 152 (2024) 3183-3196

    2024

  20. [28]

    Rome, The Hasse Norm Principle For Biquadratic Extensions J

    N. Rome, The Hasse Norm Principle For Biquadratic Extensions J. Th´ eorie des Nombres de Bordeaux 30 (2018) 947-964

    2018

  21. [29]

    J. P. Serre, A course in arithmetic . Springer-Verlag, 1973

    1973

  22. [30]

    J. P. Serre, L’invariant de Witt de la forme Tr( x2). Comment. Math. Helv. 59 (1984) 651-676

    1984

  23. [31]

    J. P. Serre, Modular forms of weight one and Galois representations, in: Algebraic number theory , 193-268. A. Fr¨ ohlich, ed. Academic Press, 1977

    1977

  24. [32]

    J. P. Serre, Topics in Galois theory . Jones and Bartlett, 1992

    1992

  25. [33]

    I. R. Shafarevich, Construction of fields of algebraic numbers with given solvable groups. Izv. Akad. Nauk. SSSR 18 (1954) 525-578; English translation in AMS translation 4 (1956) 185-237

    1954

  26. [34]

    I. R. Shafarevich, Factors of decreasing central series. Mat. Zametki 45 (1989) 114-117; English trans- lation in Math. Notes 45 (1956) 262-264. Louis M. Gaudet and Siman Wong. August 18, 2025. 37

    1989

  27. [35]

    Tenenbaum, Introduction to Analytic and Probabilistic Number Theory , 3rd ed

    G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory , 3rd ed. American Mathe- matical Society, 2015

    2015

  28. [36]

    Witt, Konstruktion von galoisschen K¨ orpern der Charakteristikp zu vorgegebener Gruppe der Ord- nung pf

    E. Witt, Konstruktion von galoisschen K¨ orpern der Charakteristikp zu vorgegebener Gruppe der Ord- nung pf . Crelle 174 (1936) 237-245

    1936

  29. [37]

    Xiong and A

    M. Xiong and A. Zaharescu, Selmer groups and Tate–Shafarevich groups for the congruent number problem. Comm. Math. Helv. 84 (1) (2009) 21-56

    2009

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.