Generalizing the supplement to Scholz's reciprocity law: an odd-prime analogue and higher-power reciprocity for the primes above 2 in $\mathbb{Q}(\sqrt{pq})$

F. Lemmermeyer's paper 'A supplement to Scholz's reciprocity law' (Acta Arith. 129 (2007), 319-324) proves, for primes $p \equiv q \equiv 1 \pmod 8$ with $(p/q)=+1$ and $(\varepsilon_p/q)=(\varepsilon_q/p)=+1$, that three statements are equivalent: the quadratic residue symbol $(\lambda_p/q)$ built from the element $\lambda_p$ generating the $h(p)$-th power of a prime ideal above $2$ in $\mathbb{Q}(\sqrt{p})$ equals $+1$; the element $\rho$ above $q$ can be chosen hyper-primary; and the ideal classes of the primes above $2$ in $\mathbb{Q}(\sqrt{pq})$ are fourth powers in its class group. Symmetry yields the reciprocity law $(\lambda_p/q)=(\lambda_q/p)$. The paper closes with two unresolved questions that constitute this problem: (1) does an analogous equivalence hold when the prime $2$ is replaced by an odd prime $\ell \neq p,q$, under suitable conditions on $p$, $q$, $\ell$; and (2) can the reciprocity law $(\lambda_p/q)=(\lambda_q/p)$ be generalized to higher power residue symbols, as Budden, Eisenmenger and Kish generalized Scholz's law itself to rational $2^n$-th power residue symbols. Both questions remain open.

Background

Let $p \equiv q \equiv 1 \pmod 4$ be rational primes with Legendre symbol $(p/q)=+1$. In the real quadratic field $\mathbb{Q}(\sqrt{p})$, with ring of integers $\mathcal{O}_p$, the fundamental unit $\varepsilon_p$ is the smallest unit greater than $1$ generating the unit group up to sign, and the class number $h(p)$ — the order of the ideal class group, which measures how far ideals are from being principal — is odd by genus theory (the part of class field theory describing how the prime factors of the discriminant force 2-torsion in the class group). Since $(p/q)=+1$, the prime $q$ splits as $q\mathcal{O}_p=\mathfrak{q}\mathfrak{q}'$. The quadratic residue symbol in a quadratic number field is defined for an ideal $\mathfrak{p}$ of odd norm by $[\alpha/\mathfrak{p}] \equiv \alpha^{(N\mathfrak{p}-1)/2} \pmod{\mathfrak{p}} \in \{\pm 1\}$, generalizing the Legendre symbol; the symbol $[\varepsilon_p/\mathfrak{q}]$ does not depend on the choice of $\mathfrak{q}$ or $\mathfrak{q}'$, so it is written $(\varepsilon_p/q)$. Scholz's reciprocity law (found by Schönemann in 1839 and rediscovered by Scholz in 1929) states that $(\varepsilon_p/q)=(\varepsilon_q/p)$; in fact both sides equal $(p/q)_4 (q/p)_4$, where the rational quartic residue symbol $(a/p)_4 := a^{(p-1)/4} \bmod p \in \{\pm 1\}$ is defined when $a$ is a quadratic residue modulo $p$. Scholz connected the symbol to the 2-class group of $F=\mathbb{Q}(\sqrt{pq})$: when $(\varepsilon_p/q)=+1$ one has $4 \mid h(pq)$.

Two notions from Hecke's theory underly the supplement. An element $\alpha$ of odd norm in the integers of a number field $K$ is called primary if it is totally positive (positive under every embedding) and $\alpha \equiv \xi^2 \pmod 4$ for some algebraic integer $\xi$; this holds exactly when the quadratic extension $K(\sqrt{\alpha})/K$ is unramified at all primes above $2$ and infinity. If $2\mathcal{O}_K=\mathfrak{l}_1^{e_1}\cdots\mathfrak{l}_r^{e_r}$, then $\alpha$ is hyper-primary if it is primary and $\alpha \equiv \xi^2 \pmod {4\mathfrak{l}_1\cdots\mathfrak{l}_r}$, i.e. exactly when every prime above $2$ splits in $K(\sqrt{\alpha})/K$. Hilbert's supplementary laws for fields of odd class number $h$ (proved in full generality by Furtwängler) say: an odd-norm ideal $\mathfrak{a}$ satisfies $(\varepsilon/\mathfrak{a})=+1$ for every unit $\varepsilon$ if and only if $\mathfrak{a}^h=(\alpha)$ for some primary $\alpha$ (First Law), and a primary ideal satisfies $(\lambda/\mathfrak{a})=+1$ for all $\lambda$ whose prime divisors lie above $2$ if and only if $\mathfrak{a}^h=(\alpha)$ for some hyper-primary $\alpha$ (Second Law).

Lemmermeyer's supplement is the Second-Law analogue of Scholz's theorem. Assume $p \equiv q \equiv 1 \pmod 8$ (so that $2$ splits, $2\mathcal{O}_p=\mathfrak{l}\mathfrak{l}'$, $2\mathcal{O}_q=\mathfrak{m}\mathfrak{m}'$), $(p/q)=+1$, and $(\varepsilon_p/q)=(\varepsilon_q/p)=+1$. Choose a totally positive $\lambda_p$ with $\mathfrak{l}^{h(p)}=(\lambda_p)$, and $\rho \in \mathcal{O}_p$ with $\mathfrak{q}^{h(q)}=(\rho)$ for a prime $\mathfrak{q}$ above $q$. Then the following are equivalent: (1) $(\lambda_p/q)=+1$; (2) $\rho$ can be chosen hyper-primary; (3) the ideal classes generated by the ideals above $2$ in $\mathbb{Q}(\sqrt{pq})$ are fourth powers in its class group. The proof uses class field theory: since $\rho$ is primary, $\mathbb{Q}(\sqrt{pq},\sqrt{\rho})$ is the unique cyclic quartic unramified extension of $F$, and $2$ splits completely in $\mathbb{Q}(\sqrt{p},\sqrt{q})$; the decomposition law then links hyper-primarity of $\rho$ to the fourth-power condition. Because statement (3) is symmetric in $p$ and $q$, one obtains the reciprocity law $(\lambda_p/q)=(\lambda_q/p)$, which also admits an elementary proof via the quadratic reciprocity law in $\mathbb{Z}[\sqrt{2}]$: writing $p=N\pi_2$, $q=N\rho_2$ with $\pi_2,\rho_2 \in \mathbb{Z}[\sqrt{2}]$, $\pi_2 \equiv \rho_2 \equiv 1 \pmod 2$, one has $(\lambda_p/q)=[\pi_2/\rho_2]=[\rho_2/\pi_2]=(\lambda_q/p)$.

Separately, Budden, Eisenmenger and Kish (2007) generalized Scholz's original law to higher powers: using the subfields $K_{2^{t-1}} \subset K_{2^t}$ of the cyclotomic field $\mathbb{Q}(\zeta_p)$ (obtained by adjoining a primitive $p$-th root of unity; its subfields correspond to the divisors of $p-1$) and a careful choice of primitive element for $K_{2^t}/K_{2^{t-1}}$, they proved reciprocity laws for the rational $2^n$-th power residue symbol $(a/p)_{2^n} := a^{(p-1)/2^n} \bmod p$, defined for primes $p \equiv 1 \pmod {2^n}$ and integers $a$ with $a^{(p-1)/2^n} \equiv \pm 1$; the octic case $2^3$ had earlier been handled by Buell and Williams. Lemmermeyer's paper closes by asking whether the supplement admits the same two extensions, and these two questions are the present problem.

Problem Statement

Both questions concern the same object — Lemmermeyer's supplement to Scholz's reciprocity law for the primes above $2$ in $\mathbb{Q}(\sqrt{pq})$, namely the equivalence of Theorem 4 (residue symbol $\Leftrightarrow$ hyper-primary element $\Leftrightarrow$ fourth-power ideal classes) together with its reciprocity law $(\lambda_p/q)=(\lambda_q/p)$ — and ask whether this supplement extends beyond its original setting.

Question 1 (odd-prime analogue). In the source's own words: the ideal class generated by a prime above $2$ in $\mathbb{Q}(\sqrt{pq})$ is a fourth power in the class group if and only if $[\pi_2/\rho_2]=+1$, where $\pi_2,\rho_2 \in \mathbb{Z}[\sqrt{2}]$ are elements $\equiv 1 \pmod 2$ with norms $p$ and $q$ — does an analogous statement hold with $2$ replaced by an odd prime $\ell \neq p,q$? The source poses the question hypothetically, "under suitable assumptions": the exact hypotheses on $p$, $q$, $\ell$ — playing the role of $p \equiv q \equiv 1 \pmod 8$, $(p/q)=+1$, $(\varepsilon_p/q)=(\varepsilon_q/p)=+1$ when $\ell=2$ — the correct $\ell$-analogue of the residue-symbol condition, and the correct power condition on the ideal classes above $\ell$ in $\mathbb{Q}(\sqrt{pq})$ are themselves part of what must be found.

Question 2 (higher powers). Can the reciprocity law $(\lambda_p/q)=(\lambda_q/p)$ be generalized to higher power residue symbols, in the way Budden, Eisenmenger and Kish generalized Scholz's law itself to rational $2^n$-th power residue symbols — the precise higher-power formulation, including its hypotheses on $p$ and $q$, being itself part of what is to be found?

The two questions are the independent open problems with which the source closes. The source poses them hypothetically ("under suitable assumptions"): the correct hypotheses on $p$, $q$, $\ell$ and the correct power condition on the ideal classes in an odd-$\ell$ analogue are themselves part of the answer.

References

1. Franz Lemmermeyer, "A supplement to Scholz's reciprocity law", Acta Arithmetica 129 (2007), no. 4, 319–324. DOI: 10.4064/aa129-4-2, https://doi.org/10.4064/aa129-4-2 (arXiv:1310.6599, https://arxiv.org/abs/1310.6599)
2. Mark Budden, Jeremiah Eisenmenger, Jonathan Kish, "A generalization of Scholz's reciprocity law", Journal de théorie des nombres de Bordeaux 19 (2007), no. 3, 583–594. DOI: 10.5802/jtnb.604, https://doi.org/10.5802/jtnb.604
3. Franz Lemmermeyer, "Scholz's reciprocity law in number fields", preprint (2021), arXiv:2101.03145, https://arxiv.org/abs/2101.03145
4. Markus Hittmeir, "Computational aspects of rational residuosity", International Journal of Number Theory 14 (2018), no. 3, 631–645. DOI: 10.1142/S1793042118500380, https://doi.org/10.1142/S1793042118500380 (arXiv:1604.07791, https://arxiv.org/abs/1604.07791)
5. Duncan A. Buell, Kenneth S. Williams, "An octic reciprocity law of Scholz type", Proceedings of the American Mathematical Society 77 (1979), no. 3, 315–318. DOI: 10.1090/S0002-9939-1979-0545588-4, https://doi.org/10.1090/S0002-9939-1979-0545588-4
6. Arnold Scholz, "Über die Lösbarkeit der Gleichung t² − Du² = −4", Mathematische Zeitschrift 39 (1934), 95–111. DOI: 10.1007/BF01201346, https://doi.org/10.1007/BF01201346
7. Kenneth S. Williams, "On Scholz's reciprocity law", Proceedings of the American Mathematical Society 64 (1977), no. 1, 45–46. DOI: 10.2307/2040978, https://doi.org/10.2307/2040978
8. Mark Budden, Alex Collins, Kristin Ellis Lea, Stephen Savioli, "Rational residuacity of primes", Involve, a Journal of Mathematics 3 (2010), no. 3, 249–257. DOI: 10.2140/involve.2010.3.249, https://doi.org/10.2140/involve.2010.3.249
9. Franz Lemmermeyer, "Octic Hilbert 2-class fields of real quadratic fields with discriminant 8p", preprint (2025), arXiv:2510.10295, https://arxiv.org/abs/2510.10295