As the title of the paper is not very clear, the intentions as due to the authors will be set out and quoted below as done in the introduction. The working-out of all this is exemplary.
“Introduction.
Several problems related to Artin’s primitive root conjecture may be viewed as instances of the following:
Index Map Problem. Let \(K\) be a number field. Let \(W_1,\dots,W_n\) be finitely generated subgroups of \(K^\times\) of positive rank. For all but finitely many primes \(\mathfrak{p}\) of \(K\), their reductions modulo \(\mathfrak{p}\) are subgroups of \(k^\times_{\mathfrak{p}}\) of finite index (\(k_{\mathfrak{p}}\) being the residue field), yielding the index map \[\mathfrak{p}\mapsto(\text{Ind}_{\mathfrak{p}} (W_1),\dots,\text{Ind}_{\mathfrak{p}}(W_n)).\] Determine whether the preimage of a subset of \(\mathbb{Z}_{>0}^n\) is infinite (respectively, it has a positive natural density). Possibly consider a finite Galois extension \(F/K\), a union of conjugacy classes \(C\subset\text{Gal}(F/K)\), and restrict the index map to the primes \(\mathfrak{p}\) unramified in \(F\) such that \(\left(\frac{F/K}{\mathfrak{p}}\right)\subset C\).
Consider the preimages of a tuple \((h_1,\dots, h_n)\in \mathbb{Z}^n_{>0}\). The original Artin’s conjecture (possibly for higher rank) corresponds to the case \(n=1\) and \(h_1=1\) because we look for a primitive root, and for a near-primitive root we can vary \(h_1\in\mathbb{Z}_{>0}\). For simultaneous primitive roots we may consider the preimage of the tuple \((1,\dots,1)\), and more generally the Schinzel-Wójcik problem concerns the constant tuples. For the two variable Artin conjecture we consider \(n=2\) and all pairs \((h_1,h_2)\) such that \(h_1\mid h_2\). For the smallest primitive root problem we let \(W_1,\dots,W_n\) be generated by the first \(n\) integers greater than one and consider the tuples such that \(h_n\) is the only entry equal to \(1\). Notice that, over \(\mathbb{Q}\), for all above-mentioned problems we could restrict to primes satisfying some congruence condition, which can be expressed by a suitable choice of \(F\) and \(C\). Also notice that, in case we do not need the Frobenius condition, we may simply take \(F= K\).
The above list of questions is surely non-exhaustive: we refer the reader to the survey by Moree on Artin’s conjecture [P. Moree, Integers 12, No. 6, 1305–1416, A13 (2012; Zbl 1271.11002)] and, to name a few, to the following papers [D. R. Heath-Brown, Q. J. Math., Oxf. II. Ser. 37, 27–38 (1986; Zbl 0586.10025); C. Hooley, J. Reine Angew. Math. 225, 209–220 (1967; Zbl 0221.10048); K. R. Matthews, Acta Arith. 29, 113–146 (1976; Zbl 0335.10007); P. Moree and P. Stevenhagen, J. Number Theory 85, No. 2, 291–304 (2000; Zbl 0966.11042); M. R. Murty et al., J. Number Theory 194, 8–29 (2019; Zbl 1437.11141); F. Pappalardi, Math. Comput. 66, No. 218, 853–868 (1997; Zbl 0883.11041); J. Wójcik, Math. Proc. Camb. Philos. Soc. 119, No. 2, 191–200 (1996; Zbl 0854.11051)].
The main results of this work address the index map problem in full generality.
Main theorem. Assume GRH: Fix a finite Galois extension \(F/K\) and a union of conjugacy classes \(C\subset\text{Gal}(F/K)\). Restrict the index map to the primes \(\mathfrak{p}\) of \(K\) unramified in \(F\) such that \(\left(F/K{\mathfrak{p}}\right)\subset C\).

(i)

[Theorem 4.1 and Remark 4.2] The preimage under the index map of any non-empty subset of \(\mathbb{Z}^n_{>0}\) is either finite or it has a positive natural density.

(ii)

[Theorems 5.4] The image of the index map is computable with an explicit finite procedure.

(iii)

[Theorem 6.2] If \(F=K\), then the following two conditions are equivalent:

–

the image of the index map contains all positive multiples of some tuple \((H_1,\dots, H_n)\);

–

for every \(i=1,\dots,n\) the rank of \(\langle W_1,\dots,W_n\rangle\) is strictly larger than the rank of \(\langle W_1,\dots, W_{i-1}, W_{i+1},\dots, W_n\rangle\). Moreover, if the above conditions hold, than a tuple \((h_1,\dots,h_n)\in \mathbb{Z}^n_{>0}\) is in the image of the index map if and only if the same holds for the tuple \[(\text{gcd}(h_1,H_1),\dots, \text{gcd}(h_n, H_n)).\]

Let \(F=K\). In Theorem 6.4 we explicitly determine the image of the index map when \(K=\mathbb{Q}\), \(n=1\) and \(W_1\) has rank \(1\). While for \(K=\mathbb{Q}\) the index map is never surjective onto \(\mathbb{Z}^n_{>0}\) if \(n\ge 2\), we prove that for any \(K\ne\mathbb{Q}\) and for any \(n\) there are groups \(W_1,\dots, W_n\) for which the index map is surjective onto \(\mathbb{Z}^n_{>0}\), see Sect. 7. Notice that in the Index Map Problem the assumption that the groups have positive rank is justified by Remarks 4.4 and 6.7.
We make use of several results of Kummer theory, which we collect in Sect. 3 (some of our statements may be new). We also prove the following result of Kummer theory, that holds for every number field \(K\ne\mathbb{Q}\): there are countably many elements of \(K^\times\) such that any finite subset of them gives rise to Kummer extensions with maximal degree [in short, for \(\alpha_1,\dots,\alpha_r\in K^\times\), the degree of \(K(\zeta_{mn},\alpha^{1/n}_1,\dots,\alpha^{1/n}_r)\) over \(K(\zeta_{mn})\) equals \(n^r\) for all positive integers \(n\) and \(m\)], see Proposition 7.1.
Our main technical tool is Theorem 4.1, which is a generalization of a celebrated result by H. W. Lenstra jun. [Invent. Math. 42, 201–224 (1977; Zbl 0362.12012)], see also [O. Järviniemi, Proc. Am. Math. Soc. 149, No. 9, 3651–3668 (2021; Zbl 1479.11171), Sect. 4] (notice that the analytic ideas remain unchanged from Hooley [loc. cit.]). Since in the proof of Theorem 4.1 we use an effective version of the Chebotarev density theorem which is conditional on GRH, most of our results on the Index Map Problem are conditional on GRH, as it is customary for problems related to Artin’s primitive root conjecture.”