Let $\theta_m \in D_1$ be a $2^m$-th root of $a + ib$, then $\overline{\theta_m}$ a $2^m$-th root of $a - ib$. Let $\gamma_1 \in G_1$ and $\gamma_2 \in G_2$ be such that $\gamma_1(\theta_m) = \zeta\theta_m$ and $\gamma_2(\overline{\theta_m}) = \overline{\zeta\theta_m}$. Since $\zeta$ and $\overline{\zeta}$ are both $2^m$-th primitive roots of unity, $\gamma_1$ and $\gamma_2$ are generators of $G_1$ and $G_2$, respectively. Moreover $\pi\gamma_1\pi(\overline{\theta_m}) = \pi\gamma_1(\theta_m) = \pi(\zeta\theta_m) = \overline{\zeta\theta_m}$, and the restriction of $\pi\gamma_1\pi$ to $D_1$ is the identity. We deduce that $\pi\gamma_1\pi = \gamma_2$, showing that $\gamma_1$ and $\gamma_2$ are conjugate in $G^+$. Since $G_1$ is cyclic, $\gamma_1^{2^{m-n}}$ is a generator of $H$. Letting $\sigma_1 = \gamma_1^{2^{m-n}}$ and $\sigma_2 = \gamma_2^{2^{m-n}}$ we conclude that $G$ (resp. $G^+$) satisfies $\mathcal{P}$ (resp. $\mathcal{P}^+$) relatively to $(\sigma_1, \sigma_2)$. $\square$

Keeping the notation as in Theorem 1.8, we conclude by giving the structure of Gal$(D_1D_2/\mathbb{Q})$ which, by Proposition 4.6, satisfies $\mathcal{P}^+$ relatively to $(\sigma_1, \sigma_2)$.

**Proposition 4.13.** We have an isomorphism

$$ \mathrm{Gal}(D_1D_2/\mathbb{Q}) \simeq ((\mathbb{Z}/2^m\mathbb{Z} \times \mathbb{Z}/2^m\mathbb{Z}) \times \mathbb{Z}/2\mathbb{Z}) \times \mathbb{Z}/2^{m-2}\mathbb{Z}. $$

*Proof.* We keep the notation as in the proof of Theorem 1.8.

We have $\mathrm{Gal}(K/\mathbb{Q}) = \langle\pi, \tau_0\rangle$ where $\tau_0(\zeta) = \zeta^5$. Let $\tau_1$ be an extension of $\tau_0$ to $D_1D_2$. The coset $\tau_1\mathrm{Gal}(D_1D_2/K)$ contains exactly the automorphisms of $D_1D_2$ that restrict to $\tau_0$.

We claim that there exists $\tau \in G^+$ of order $2^{m-2}$ extending $\tau_0$. Indeed, as $\tau_0$ fixes $\mathbb{Q}(i)$ (recall that $\tau_0(i) = i$), $\tau_1$ permutes the roots of $X^{2^m} - (a + ib)$ (resp. $X^{2^m} - (a - ib)$). Moreover there exist $d_1, d_2 \in \mathbb{Z}$ such that $\tau_1(\theta_m) = \zeta^{d_1}\theta_m$ and $\tau_1(\overline{\theta_m}) = \overline{\zeta^{d_2}\theta_m}$. Also since 5 is invertible modulo $2^m$, there exist $r_1, r_2 \in \mathbb{Z}$ such that $5r_2 + d_2 \equiv 5r_1 + d_1 \equiv 0 \pmod{2^m}$. Set $\tau = \tau_1\gamma_1^{r_1}\gamma_2^{-r_2}$. By construction, $\tau(\theta_m) = \theta_m$ and $\tau(\overline{\theta_m}) = \overline{\theta_m}$. In particular we have $\tau^{2^{m-2}} = \mathrm{Id}_{D_1D_2}$. However $\mathrm{ord}(\tau) \ge 2^{m-2}$ since $\tau$ extends $\tau_0$, and therefore $\mathrm{ord}(\tau) = 2^{m-2}$.

We have $\mathrm{Gal}(D_1D_2/\mathbb{Q}) = \langle\gamma_1, \gamma_2, \pi, \tau\rangle$, moreover, as $D_1D_2 = \mathbb{Q}(\zeta, \theta_m, \overline{\theta_m})$ one easily checks the equalities

$$ \tau\pi\tau^{-1} = \pi, \quad \tau\gamma_1\tau^{-1} = \gamma_1^5, \quad \tau\gamma_2\tau^{-1} = \gamma_2^5. $$

Therefore one has $G^+ \triangleleft \mathrm{Gal}(D_1D_2/\mathbb{Q})$ and thus $\langle\gamma_1, \gamma_2, \pi, \tau\rangle = \langle\gamma_1, \gamma_2, \pi\rangle \cdot \langle\tau\rangle$. However, $G^+ = \langle\gamma_1, \gamma_2, \pi\rangle \simeq (\mathbb{Z}/2^m\mathbb{Z} \times \mathbb{Z}/2^m\mathbb{Z}) \times \mathbb{Z}/2\mathbb{Z}$. Therefore, for cardinality reasons (see Lemma 3.14), $\langle\gamma_1, \gamma_2, \pi\rangle \cap \langle\tau\rangle = \{1\}$. Then the product $\langle\gamma_1, \gamma_2, \pi\rangle \cdot \langle\tau\rangle$ is semi-direct which completes the proof. $\square$

*Remark* 4.4. In the case $m = 2$, we recover an explicit solution to the inverse Galois problem for the group $(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}) \times \mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Q}$. Recall that by Proposition 4.4, this is the smallest group containing $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ that satisfies $\mathcal{P}^+$.

**Acknowledgments.** I would like to thank Florent Jouve for his constant support and patience, providing invaluable guidance and insights throughout the work that led to this paper. I would also like to thank Daniel Fiorilli for several discussions that significantly enriched my research.

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[5] P. Chebyshev, *Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveau théorème relatif aux nombres premiers contenus dans les formes $4n + 1$ et $4n + 3$*, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg (1853), 208.

[6] L. Devin, *Chebyshev's bias for analytic L-functions*, Math. Proc. Cambridge Philos. Soc. **169** (2020), no. 1, 103–140.

[7] W. Feit and G. M. Seitz, *On finite rational groups and related topics*, Illinois J. Math. **33** (1989), no. 1, 103–131. MR0974014

[8] D. Fiorilli and F. Jouve, *Distribution of Frobenius elements in families of Galois extensions*, forthcoming, J. Inst. Math. Jussieu, available at https://doi.org/10.1017/S1474748023000154.

[9] __________, *Unconditional Chebyshev biases in number fields*, J. Éc. polytech. Math. **9** (2022), 671–679.

[10] M. Hall Jr., *The theory of groups*, The Macmillan Company, New York, 1959.

[11] J. Kaczorowski, *On the distribution of primes (mod 4)*, Analysis **15** (1995), no. 2, 159–171.

[12] S. Lang, *Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.

[13] N. Ng, *Limiting distributions and zeros of Artin L-functions* (2000), available at https://www.cs.uleth.ca/~nathanng/RESEARCH/phd.thesis.pdf.

[14] M. Rubinstein and P. Sarnak, *Chebyshev's bias*, Experiment. Math. **3** (1994), no. 3, 173–197.