Abstract
Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH).
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1 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 \(W_1, \ldots , W_n\) 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 \(k_\mathfrak {p}^\times \) of finite index (\(k_\mathfrak {p}\) being the residue field), yielding the index map
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 \(C \subset Gal (F/K)\), 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 \((h_1, \ldots , h_n) \in {\mathbb {Z}}_{>0}^n\). 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 nearprimitive root we can vary \(h_1\in {\mathbb {Z}}_{>0}\). For simultaneous primitive roots we may consider the preimage of the tuple \((1, \ldots , 1)\), 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 \(W_1,\ldots , W_n\) 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 abovementioned 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 nonexhaustive: we refer the reader to the survey by Moree on Artin’s Conjecture [11] and, to name a few, to the following papers [3, 5, 10, 12,13,14, 21].
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 \(C \subset Gal (F/K)\). Restrict the index map to the primes \({\mathfrak {p}}\) of K unramified in F such that \(\left( \frac{F/K}{{\mathfrak {p}}}\right) \subset C\).

(i)
[Theorem 4.1 and Remark 4.2] The preimage under the index map of any nonempty subset of \({\mathbb {Z}}_{>0}^n\) 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 \((H_1,\ldots , H_n)\);

for every \(i=1,\ldots , n\) the rank of \(\langle W_1,\ldots , W_n\rangle \) is strictly larger than the rank of \(\langle W_1,\ldots ,W_{i1}, W_{i+1}, \ldots , W_n\rangle \).
Moreover, if the above conditions hold, then a tuple \((h_1,\ldots , h_n)\in {\mathbb {Z}}_{>0}^n\) is in the image of the index map if and only if the same holds for the tuple
$$\begin{aligned} (\gcd (h_1, H_1),\ldots , \gcd (h_n, H_n)). \end{aligned}$$ 
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 \({\mathbb {Z}}_{>0}^n\) if \(n \geqslant 2\), we prove that for any \(K\ne {\mathbb {Q}}\) and for any n there are groups \(W_1,\ldots , W_n\) for which the index map is surjective onto \({\mathbb {Z}}_{>0}^n\), 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 \(\alpha _1,\ldots , \alpha _r\in K^\times \) , the degree of \(K(\zeta _{mn}, \alpha _1^{1/n}, \ldots , \alpha _r^{1/n})\) 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 Lenstra [9], see also [7, Sect. 4] (notice that the analytic ideas remain unchanged from Hooley [5]). 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.
2 Notations for the paper
General notation. In this paper K is a number field, F/K a finite Galois extension of K with Galois group \(Gal (F/K)\), and \(C \subset Gal (F/K)\) is a union of conjugacy classes. The multiplicative group of K is denoted by \(K^{\times }\), the ring of integers by \(O_K\), and we fix some algebraic closure \({\bar{K}}\) containing F. We call \(\tau \) the order of the torsion subgroup of \(K^\times \). If n is a positive integer, then we write \(\zeta _n\) for a root of unity of order n (such that for all positive integers m we have \(\zeta _{mn}^m=\zeta _n\)). We also use the notation \(K(\zeta _{\infty })\) and \(K(\zeta _{\ell ^\infty })\), in the obvious way, to define infinite cyclotomic extensions of K. We call \(\Omega \) the smallest positive integer such that \(K\cap \mathbb Q(\zeta _\infty )\subset {\mathbb {Q}}(\zeta _\Omega )\). The letters \(p, q, \ell \) are reserved for primes (of \({\mathbb {Z}}\)), and \({\mathfrak {p}}, {\mathfrak {q}}\) for primes of K i.e. nonzero prime ideals of \(O_K\). The ideal norm of \({\mathfrak {p}}\) is \(N{\mathfrak {p}}\), the residue field is \(k_{\mathfrak {p}}\), and the Artin symbol is the conjugacy class \(\left( \frac{F/K}{{\mathfrak {p}}}\right) \).
Tuples. We write \(I=\{1,\ldots , n\}\) and hence we denote e.g. by \(h_I\) the tuple \((h_1,\ldots , h_n)\). If \(h_I\) and \(H_I\) are in \({\mathbb {Z}}_{\geqslant 0}^n\), then \(h_I\) is smaller than or equal to \(H_I\) if \(h_i\leqslant H_i\) holds for every \(i\in I\), and we define \(\min (h_I, H_I)\) as the tuple with entries \(\min (h_i, H_i)\). Moreover, we call \(\oplus h_I\) the set consisting of the tuples larger than or equal to \(h_I\). If \(h_I\) and \(H_I\) are in \(\mathbb Z_{> 0}^n\), then \(h_I\) divides \(H_I\) if \(h_i\mid H_i\) holds for every \(i\in I\), and \(\gcd (h_I, H_I)\) is the tuple with entries \(\gcd (h_i, H_i)\). Moreover, we call \(\otimes h_I\) the set consisting of the multiples of \(h_I\), and the \(\ell \)adic valuation \(v_\ell (h_I)\) is the tuple with entries \(v_\ell (h_i)\).
Groups. The symbol W stands for a finitely generated subgroup of \(K^\times \) of positive rank (by which we mean the rank of W modulo its torsion subgroup \(W_{{{\,\textrm{tors}\,}}}\)). For all but finitely many \({\mathfrak {p}}\) the reduction \((W \bmod {\mathfrak {p}})\) is a welldefined subgroup of \(k_{\mathfrak {p}}^\times \) and we consider its index \(Ind _{{\mathfrak {p}}}W\), which we also write as \(Ind _{{\mathfrak {p}}}\alpha \) if \(W=\langle \alpha \rangle \). If \(n\in {\mathbb {Z}}_{>0}\), then \(W^{1/n}\subset {\bar{K}}^\times \) consists of all nth roots of all elements of W (we also consider \(S^{1/n}\) for \(S\subset K^\times \)), and we similarly define \(W^{1/\infty }\) and \(W^{1/{\ell ^\infty }}\).
Let \(I=\{1,\ldots , n\}\) and for \(i\in I\) consider subgroups \(W_i\) of \(K^\times \) of positive rank. Call \(W:=\langle W_i \mid i\in I\rangle \) and \(W_{\ne i}:=\langle W_j \mid j\in I, j\ne i \rangle \).
Separated groups. We call \(W_1,\ldots , W_n\) separated if for all \(i\in I\) we have
The index map. For all but finitely many primes \({\mathfrak {p}}\) of K we can define the index map
Its \(\ell \)adic valuation \(\Psi _\ell \) is the composition of \(\Psi \) with the \(\ell \)adic valuation \({\mathbb {Z}}_{>0}^n\rightarrow {\mathbb {Z}}_{\geqslant 0}^n\). We similarly define its Qadic valuation \(\Psi _Q\) for any positive squarefree integer Q, where the Qadic valuation \(v_Q(z)\) of \(z\in {\mathbb {Z}}_{\ne 0}\) is the tuple consisting of the \(\ell \)adic valuation \(v_\ell (z)\) for the primes \(\ell \mid Q\).
Convention. Starting from Theorem 4.1, up to excluding finitely many primes of K, we restrict \(\Psi \) to the primes that are not ramified over \({\mathbb {Q}}\), do not ramify in F, and do not lie over a prime divisor of \(\tau \).
3 Results from Kummer theory
Proposition 3.1
The following holds:

(i)
There exists a positive integer z (depending on K and W, computable with an explicit finite procedure) such that for all positive integers t we have
$$\begin{aligned} \left[ K\left( \zeta _\infty , W^{1/zt}\right) : K\left( \zeta _\infty , W^{1/z}\right) \right] =t^{rank W}. \end{aligned}$$ 
(ii)
There exists a finite Galois extension \(K_W/K\) (depending on K and W, computable with an explicit finite procedure) such that for all coprime positive integers m, t we have
$$\begin{aligned} K\left( \zeta _{m}, W^{1/m}\right) \cap K\left( \zeta _{t}, W^{1/t}\right) \subset K_W. \end{aligned}$$ 
(iii)
There exists a positive integer c (depending on K and W, computable with an explicit finite procedure) such that for all positive integers m, t such that m is coprime to both t and c we have
$$\begin{aligned}{} & {} K\left( \zeta _m, W^{1/m}\right) \cap K\left( \zeta _{t}, W^{1/t}\right) =K \\{} & {} \text {and}\quad [K\left( \zeta _m, W^{1/m}\right) : K]=\varphi (m)m^{rank W}. \end{aligned}$$We could additionally require that \(K\left( \zeta _m, W^{1/m}\right) \cap F=K\) (in this case c also depends on F). We could additionally require that c is divisible by some given positive integer, or we may replace c by its squarefree part.
Proof
For (i) we may suppose that W is torsionfree. By [17, Theorems 1.1 and 1.2] we can take z to be a positive integer maximizing the integer ratio
We may also suppose that \(K\cap {\mathbb {Q}}(\zeta _\infty )\subset K(\zeta _z)\), so that for (ii) and (iii) we may take \(K_W=K\left( \zeta _z, W^{1/z}\right) \) and \(c=z\). For the additional remarks in (iii) notice that, since we may replace c by a multiple, we may suppose \(\Omega \mid c\) and \(F\cap K\left( \zeta _\infty , W^{1/\infty }\right) \subset K\left( \zeta _c, W^{1/c}\right) \). \(\square \)
In the following result we introduce the terminology of separated (respectively, \(\ell \)separated) Kummer extensions.
Proposition 3.2
Suppose that \(W_1,\ldots , W_n\) are separated.

(i)
There exists \(h_I\in {\mathbb {Z}}_{>0}^n\) such that the \(W_i\)’s have separated Kummer extensions w.r.t. \(h_I\). By this we mean that the fields
$$\begin{aligned} K\left( \zeta _\infty , W_1^{1/H_1}, \ldots , W_n^{1/H_n}\right) \quad \text {for}\quad H_I\in \otimes h_I \end{aligned}$$are pairwise distinct. Equivalently, by varying \(i \in I\) and \(H_i\in h_i {\mathbb {Z}}\cap {\mathbb {Z}}_{>0},\) the fields \(K(\zeta _\infty , W_i^{1/H_i}, W_{\ne i}^{1/\infty })\) are pairwise distinct.

(ii)
For every \(\ell \) there exists \(e_I\in \mathbb Z_{\geqslant 0}^n\) such that the \(W_i\)’s have \(\ell \)separated Kummer extensions w.r.t. \(e_I\). By this we mean that the fields
$$\begin{aligned} K\left( \zeta _{\ell ^\infty }, W_1^{1/\ell ^{E_1}}, \ldots , W_n^{1/\ell ^{E_n}}\right) \quad \text {for}\quad E_I\in \oplus e_I \end{aligned}$$are pairwise distinct. Equivalently, by varying \(i \in I\) and \(E_i\in {\mathbb {Z}}_{\geqslant e_i},\) the fields \(K(\zeta _{\ell ^\infty }, W_i^{1/\ell ^{E_i}}, W_{\ne i}^{1/\ell ^{\infty }})\) are pairwise distinct.

(iii)
Consider the set \(Sep\subset {\mathbb {Z}}_{>0}^n\) (respectively, \(Sep_\ell \subseteq {\mathbb {Z}}_{\geqslant 0}^n)\) consisting of the tuples with respect to which the \(W_i\)’s have separated (respectively, \(\ell \)separated) Kummer extensions. The set Sep and all sets \(Sep_\ell \) are computable at once with an explicit finite procedure. There exist \(h_I\in {\mathbb {Z}}_{>0}^n\) and \(e_{\ell ,I}\in \mathbb Z_{\geqslant 0}^n\) such that \(Sep=\otimes h_I,\) \(Sep_\ell =\oplus e_{\ell ,I},\) and for every \(i\in I\) we have \(0\leqslant v_\ell (h_i) e_{\ell , i} \leqslant v_\ell (\tau )\) and
$$\begin{aligned} v_\ell (h_i)&= \max \left\{ z\,: \; K\left( \zeta _{\infty }, W_{i}^{1/\ell ^{z}}\right) \subset K\left( \zeta _{\infty }, W_{\ne i}^{1/\ell ^{\infty }}, W_i^{1/\ell ^{z1}}\right) \right\} ,\\ e_{\ell , i}&= \max \left\{ z\,: \; K\left( \zeta _{\ell ^\infty }, W_{i}^{1/\ell ^{z}}\right) \subset K\left( \zeta _{\ell ^\infty }, W_{\ne i}^{1/\ell ^{\infty }}, W_i^{1/\ell ^{z1}}\right) \right\} . \end{aligned}$$
By Proposition 3.2(i), for every \(H_I\in \otimes h_I\) the degree
is a multiple of \(\prod _{i\in I} H_i/h_i\) (stepwise, multiplying one parameter by a prime \(\ell \) gives a nontrivial Kummer extension whose degree is a multiple of \(\ell \)). Similarly, by Proposition 3.2(ii) for every \(E_I\in \oplus e_I\) the \(\ell \)adic valuation of the degree
is at least \(\sum _{i\in I} (E_ie_i)\).
Proof
To prove the first equivalence (the second is analogous), consider tuples \(H_I, H_I'\in \otimes h_I\). If we have \(K(\zeta _\infty , W_1^{1/H_1}, \ldots , W_n^{1/H_n})=K(\zeta _\infty , W_1^{1/H'_1}, \ldots , W_n^{1/H'_n})\), then clearly for every i we have \(K(\zeta _\infty , W_i^{1/H_i}, W_{\ne i}^{1/\infty })=K(\zeta _\infty , W_i^{1/H'_i}, W_{\ne i}^{1/\infty })\). For the converse: if the latter equality holds for some i and for some \(H_i, H_i'\), then for \(j\ne i\) we can set \(H_j=H_j'=Z\) for some (sufficiently divisible) positive integer Z so that the former equality holds.
Now we prove (i), the proof of (ii) being analogous. Let z be as in Proposition 3.1(i) for W, thus for every positive multiple Z of z we have
By assumption \(rank W>rank W_{\ne i}\) hence for every distinct positive divisors \(t,t'\) of Z/z we have \(K(\zeta _\infty , W_i^{1/tz}, W_{\ne i}^{1/Z})\ne K(\zeta _\infty , W_i^{1/t'z}, W_{\ne i}^{1/Z})\) Indeed, we have
As we can freely choose Z, we may replace it by \(\infty \).
We are left to prove (iii). Notice that \(m_I \in Sep\) (respectively, \(Sep_\ell \)) implies \(\otimes m_I\subset Sep\) (respectively, \(\oplus m_I\subset Sep_\ell \)). Call \(v_{\ell , i}\) and \(e_{\ell , i}\) the two maxima from (iii), and set \(h_i:=\prod _\ell \ell ^{v_{\ell , i}}\), thus \(Sep \subset \otimes h_I\) and \(Sep_\ell \subset \oplus e_{\ell , I}\). These inclusions are equalities because by Kummer theory \(h_I\in Sep\) and \(e_{\ell , I}\in Sep_\ell \). Clearly \(v_\ell (h_i) e_{\ell , i}\) is nonnegative, and it is at most \(v_\ell (\tau )\) by Schinzel’s Theorem for abelian radical extensions [18, Theorem 2] (with the notation of [17], we have to consider the \(\ell \)adelic failure for W).
By the theory developed in [1, 15] we may compute \(e_{\ell , I}\) for all \(\ell \), which is the zero tuple except for finitely many computable primes \(\ell \) . We are left to compute \(v_\ell (h_I)\) for \(\ell \mid \tau \). Set \(t_\ell :=v_\ell (\tau )+\max _i(e_{\ell , i})\), and let \(T_\ell \) be the product of \(2\ell ^{v_\ell (\tau )}\) and the primes \(q\equiv 1 \pmod \ell \) such that \(\ell \not \mid v_{\mathfrak {q}} (w)\) holds for some \(w\in W\) and for some prime \({\mathfrak {q}}\) of K lying over q. We conclude because by [4, Lemma C.1.7 and its proof] we have
\(\square \)
With the notation of Proposition 3.2 we could have \(e_{\ell , i}< v_\ell (h_i)\): indeed, for \(K={\mathbb {Q}}\) and \(W=\langle 5 \rangle \) we have \(Sep_2={\mathbb {Z}}_{\geqslant 0}\) and \(Sep=2\, {\mathbb {Z}}_{>0}\).
Lemma 3.3
Let \(\alpha _1, \ldots , \alpha _r \in K^{\times }\) be multiplicatively independent. There exist a positive integer N and \(S \subset ({\mathbb {Z}}/N{\mathbb {Z}})^{r+1}\) such that for \((z_0, \ldots , z_r)\in {\mathbb {Z}}_{>0}^{r+1}\) with \(lcm (z_0, \ldots , z_r)= z_0\) the following holds: given \((x_0, \ldots , x_r)\in {\mathbb {Z}}_{>0}^{r+1},\) there is
such that \(\sigma _F \in C\), \(\sigma \left( \zeta _{z_0}\right) = \zeta _{z_0}^{x_0},\) and \(\sigma \left( \alpha _i^{1/z_i}\right) = \zeta _{z_i}^{x_i}\alpha _i^{1/z_i}\) for all \(i=1,\ldots , r\) if and only if \(\gcd (x_0, z_0) = 1\) and \((x_0 \bmod {N}, \ldots , x_r \bmod {N}) \in S\). The integer N only depends on K, F and \(\langle \alpha _1, \ldots , \alpha _r\rangle ,\) and N, S are computable with an explicit finite procedure. Supposing that the prime divisors of \(z_0, \ldots , z_r\) belong to some set D, we may remove from N all prime factors not in D.
Proof
The condition \(\gcd (x_0, z_0) = 1\) is clearly necessary, so assume that it holds. Calling \(W=\langle \alpha _1, \ldots , \alpha _r\rangle \), by Proposition 3.1 there is some positive integer N such that \(F\cap K(\zeta _\infty , W^{1/\infty })\subset K(\zeta _N, W^{1/N})\) and such that for every positive multiple M of N we have
Thus we may replace \(z_i\) by \(\gcd (z_i,N)\) and work in \(Gal (K(\zeta _N, W^{1/N})/K)\): if one finds a suitable automorphism of \(K(\zeta _N, W^{1/N})\), then one may extend it freely to \(K(\zeta _M, W^{1/M})\) as the degree of the extension above is maximal. \(\square \)
Remark 3.4
By [1, Definitions 5 and 10], elements \(\alpha _1,\ldots , \alpha _r\in K^\times \) must be strongly \(\ell \)independent if for all \(i=1,\ldots , r\) there is some prime \({\mathfrak {q}}_i\) of K such that \(\ell \not \mid v_{\mathfrak q_i}(\alpha _i)\) and \(\ell \mid v_{{\mathfrak {q}}_i}(\alpha _j)\) for all \(j\ne i\). Indeed, if \(\alpha :=\prod \alpha _{i}^{z_i}\) holds for some integers \(z_i\) and w.l.o.g. \(\ell \not \mid z_1\), then \(\ell \not \mid v_{{\mathfrak {q}}_1}(\alpha )\) hence \(\alpha \) is strongly \(\ell \)indivisible. Similarly, if the \(\alpha _i\)’s are not units and their norms \(N_{K/{\mathbb {Q}}}(\alpha _i)\) are pairwise coprime and not of the form \(\pm 1\) times an \(\ell \)th power, then they are an \(\ell \)good basis (as in [1, Theorem 14]) for \(\langle \alpha _1,\ldots , \alpha _r \rangle \).
3.1 Kummer extensions contained in cyclotomic extensions
Consider a prime power \(\ell ^e\mid \tau \) and a cyclic Kummer extension L/K of degree \(\ell ^e\) with \(L\subset K(\zeta _\infty )\). Call \(K'\) the largest subfield of \(K\cap {\mathbb {Q}}(\zeta _\infty )\) of degree a power of \(\ell \) (over \({\mathbb {Q}}\)). We then have \(L=L'K\) for some unique \(K'\subset L'\subset {\mathbb {Q}}(\zeta _\infty )\) such that \([L':{\mathbb {Q}}]\) is a power of \(\ell \) (and hence \([L': K']=\ell ^e\)). We have \(L' \subset {\mathbb {Q}}(\zeta _n)\) for some positive integer n of the form
where q denotes a prime. Indeed, if \(L' \subset {\mathbb {Q}}(\zeta _n)\) and q is a prime divisor of n with \(q \not \equiv 0, 1 \pmod {\ell }\), then \(L' \subset {\mathbb {Q}}(\zeta _{n/q})\) and hence we may replace n by n/q; the same holds if \(q \equiv 1 \pmod {\ell }\) and \(q^2 \mid n\). Notice that the extension \(L'/K'\) corresponds to a cyclic quotient of order \(\ell ^e\) of \(G:=Gal ({\mathbb {Q}}(\zeta _n)/K')\).
Proposition 3.5
Replace G by its largest quotient of exponent \(\ell ^e\) and suppose that \(G \simeq \prod _{i} {\mathbb {Z}}/\ell ^{e_i}{\mathbb {Z}}\). The ith factor corresponds to a cyclic Kummer extension \(K'(\gamma _i)/K'\) of degree \(\ell ^{e_i}\) for some \(\gamma _i\in {\mathbb {Q}}(\zeta _n)\) such that \(\gamma _i^{\ell ^{e_i}}\in K'\).

(i)
We have \(L'=K'(\gamma )\) for some \(\gamma :=\prod _{i} \gamma _i^{y_i},\) where \(0\leqslant y_i < \ell ^{e_i}\) and for some i we have \(e_i=e\) and \(\ell \not \mid y_i\).

(ii)
For every \(q\mid n\) such that \(q\not \mid \ell \Omega \) fix an algebraic integer \(g_q\in {\mathbb {Q}}(\zeta _{\ell ^e q})\) such that \(g_q^{\ell ^e}\in {\mathbb {Q}}(\zeta _{\ell ^e})\) and such that
$$\begin{aligned} v_\ell ([{\mathbb {Q}}(\zeta _{\ell ^e}, g_q):\mathbb Q(\zeta _{\ell ^e})])=\varepsilon _q\quad \text {where}\quad \varepsilon _q:=\min (e, v_\ell (q1)). \end{aligned}$$Notice that \(g_q\) depends on \(q,\ell ^e\) but not on K and L. For every \(q\mid n\) we may fix \(\gamma _i\) as follows: if \(q\not \mid \ell \Omega ,\) then \(\gamma _i^{\ell ^{e_i}}=g_q^{\ell ^{\varepsilon _q}},\) else \(\gamma _i\) is an algebraic integer in \({\mathbb {Q}}(\zeta _{\ell ^e\Omega })\). Then \(\ell \not \mid v_{\mathfrak {p}} (\gamma _i^{\ell ^{e_i}})\) holds for a prime \({\mathfrak {p}}\) of K only if \({\mathfrak {p}}\) lies over q or over a divisor of \(\ell \Omega \) respectively, and in the former case there is such \({\mathfrak {p}}\).
Proof
Since the extensions \(K'(\gamma _i)/K'\) are linearly disjoint, \(\prod _{i} \gamma _i^{z_i}\in K'\) implies \(\ell ^{e_i}\mid z_i\) for every i. Let \(0\leqslant y_i,Y_i < \ell ^{e_i}\) and suppose that for some \(i_1,i_2\) we have \(e_{i_1}=e_{i_2}=e\) and \(\ell \not \mid y_{i_1}, Y_{i_2}\). Then \(\prod _{i} \gamma _i^{y_i}\) and \(\prod _{i} \gamma _i^{Y_i}\) generate the same extension if and only if \(Y_i\equiv ty_i \pmod {\ell ^{e_i}}\) holds for every i and for some t coprime to \(\ell \), in other words if \(y_I\) and \(Y_I\) generate the same cyclic subgroup of G of order \(\ell ^e\). By Pontryagin duality the number of those subgroups is the same as the number of cyclic quotients of G of order \(\ell ^e\) and hence (i) follows. For the first assertion of (ii) it suffices to choose the isomorphism of G respecting the decomposition \(Gal (\mathbb Q(\zeta _n)/K')=Gal ({\mathbb {Q}}(\zeta _{n_1})/K')\times Gal (\mathbb Q(\zeta _{n_2})/{\mathbb {Q}})\), where \(n=n_1n_2\) and \(n_2\) consists of all prime factors \(q\not \mid \ell \Omega \). We then conclude by a result on valuations [4, Lemma C.1.7 and its proof]. \(\square \)
Example 3.6
As an aside remark, notice that L may not be contained in the compositum of K, \({\mathbb {Q}}(\zeta _{\ell ^\infty })\), and for any prime q subextensions of \({\mathbb {Q}}(\zeta _q)/K\cap \mathbb Q(\zeta _q)\) of degree dividing \(\ell ^e\): consider \(\ell ^e=2\), \(K={\mathbb {Q}}(\sqrt{65})\), and the quartic cyclic field \(L=\mathbb Q(\sqrt{65+8\sqrt{65}})\subset {\mathbb {Q}}(\zeta _{65})\).
4 The Master Theorem on Artin type problems
For \(h_I\in {\mathbb {Z}}_{>0}^n\), \(h:=lcm (h_1,\ldots , h_n)\), and a positive squarefree integer Q, we write
and we let \(C_Q \subset Gal \big (FK_Q/K_1\big )\) consist of the automorphisms whose restriction to F lies in C, and which, for any choice of \(q\mid Q\) and \(i\in I\), is not the identity on \(K(\zeta _{qh_i}, W_i^{1/qh_i})\). For every set \({\mathcal {S}}\) of primes of K we write \({\mathcal {S}}(x):= \{{\mathfrak {p}}\in \mathcal S : N{\mathfrak {p}} \le x\}\) and
The density of \({\mathcal {S}}\) is defined by
where it is possible that the limit does not exist.
Theorem 4.1
Assume GRH. Fix \(h_I\in {\mathbb {Z}}_{>0}^n,\) and let S be the set of primes \({\mathfrak {p}}\) in the domain of \(\Psi \) (restricting \(\Psi \) to the primes of K that are not ramified over \({\mathbb {Q}},\) do not ramify in F, and do not lie over a prime divisor of \(\tau )\) for which
The set S has a natural density d(S), which is 0 if and only if \(S=\emptyset \) if and only if \(C_Q = \emptyset \) for some positive squarefree integer Q. Writing \(Q_t:=\prod _{q\leqslant t} q,\) we have
The property \(\Psi (\mathfrak {p})=h_I\) means that \({\mathfrak {p}}\) splits in \(K(\zeta _{h_i}, W_i^{1/h_i})\) and it does not split in \(K(\zeta _{qh_i}, W_i^{1/qh_i})\), for any choice of \(i\in I\) and of a prime q. Requiring the above conditions only for \(q\leqslant t\), we obtain a larger set with natural density \(d_{Q_t}\). Theorem 4.1 says that these larger sets are good approximations of S for t large.
Proof
For Q a squarefree positive integer, we have \(S\subset S_Q\), where
By the Chebotarev density theorem the natural density of \(S_Q\) is \(d_Q:={C_{Q}}/{[FK_{Q} : K]}\). We have \(d_Q=0\) if and only if \(C_Q=\emptyset \), and these conditions imply \(S=\emptyset \).
We claim that for any positive integer t we have
where the implied constants depend on \(n, K, F, W, h_I\). We deduce that d(S) exists and (as \(t\mapsto d_{Q_t}\) is nonincreasing) it equals \(\inf \{d_{Q_t}\}\).
Let \(Q_0\) be the squarefree part of the constant c from Proposition 3.1(iii), assuming \(n! h\mid c\) and the additional condition involving F. We suppose \(C_{Q_0} \ne \emptyset \) and prove \(\inf \{d_{Q_t}\}>0\). Let \(L_q:=K(\zeta _q, W^{1/q})\) and \(L_{q,i}:=K(\zeta _q, W_i^{1/q})\). Since \([L_{q,i}:K]\geqslant (q1)q\) and by linear disjointness, for \(t\geqslant Q_0\) we have
The product over primes \(q > n\) is strictly positive: It suffices to show that the product over \(n< q < y\) is bounded from below independently of y. Taking the logarithm of the partial products and using the bound \(\log (1  \alpha ) \geqslant 2\alpha \) (valid for \(0 \leqslant \alpha \leqslant 1/2\)) we have
which is bounded from below by \(\exp (2n\zeta (2)) = \exp (\pi ^2 n/3) > 0\). Hence \(d_{Q_t}\) is bounded from below by a positive constant independent of t, as \(d_{Q_0} > 0\).
We are left to prove the claim (1). For t large enough, every \(q>t\) satisfies \(q\not \mid h\) and \([L_{q,i}: K] \geqslant (q1)q\). Setting \(f_1(x) := \sqrt{x}/(\log x)^{100}\) and \(f_2(x) := \sqrt{x}(\log x)^{100}\), consider the intervals
and the corresponding sets
Any \({\mathfrak {p}}\in S_{Q_t}\setminus S\) splits in some \(L_{q,i}\) with \(q>t\), and \(N{\mathfrak {p}} \leqslant x\) implies \(q\leqslant x\). Thus \(S_{Q_t}(x) S(x) = O\left( \left \Gamma _1\right + \left \Gamma _2\right + \left \Gamma _3\right \right) \). We clearly have
By the effective Chebotarev density theorem under GRH [8, 19] we get
Moreover, considering separately the primes of K of degree greater than 1 we have
and hence by the Brun–Titchmarsh inequality
where the last equality follows e.g. by decomposing \(I_2\) into dyadic intervals of the form [y, 2y], and bounding the contribution of such intervals by the prime number theorem and the estimate \(1/(q1) = O(1/y)\) for \(q \in [y, 2y]\).
Fixing some \(w_i\in W_i\setminus K^\times _{{{\,\textrm{tors}\,}}}\) we have
Thus \(\mathfrak {p}\in \Gamma _3\) implies \(w_i^{(N{\mathfrak {p}}  1)/q} \equiv 1 \pmod {{\mathfrak {p}}}\) for some \(i\in I\), \(q \in I_3\). So we have
(the implied constant also depends on \(w_i\)) and hence \(\left \Gamma _3\right = o(x/\log x)\). \(\square \)
Remark 4.2
Assuming GRH, d(S) also exists if we replace \(h_I\) by some nonempty set \(H\subset {\mathbb {Z}}_{>0}^n\), and we have
For the existence of d(S(H)), let \(K_C\) denote the primes \({\mathfrak {p}}\) of K with \(\left( \frac{F/K}{{\mathfrak {p}}}\right) \subset C\), let \(d^\pm \) denote the upper/lower density and let \(B(r) := \{1, \ldots , r\}^n\). We have
by Theorem 4.1 and because \(d^(\Psi ^{1}B(r))\rightarrow 1\). The last claim holds since it is wellknown that under GRH for \(a\in {\mathbb {Q}}\setminus \{0, \pm 1\}\) the density of the primes p such that \(Ind _p(a) \leqslant c\) exists and tends to 1 as \(c \rightarrow \infty \), and the same holds for number fields following Hooley’s proof [5, Theorem 4.1, especially (3)], see also [2, Theorem 4] or [20, Theorem 5.1].
Finally, (2) follows from the observations above because, as \(r \rightarrow \infty \), we have
In the following result we work with \(\Psi _T\) for any squarefree integer \(T\geqslant 2\). An image for this map is a tuple of images of \(\Psi _\ell \) for \(\ell \mid T\) prime. To ease notation we describe an element of \(\Psi _T\) as a matrix whose entries are nonnegative integers \(e_{i,\ell }\).
Corollary 4.3
Assume GRH for (i).

(i)
Let \(h_I\in {\mathbb {Z}}_{>0}^n\) and \(h:=lcm (h_1,\ldots , h_n)\). We have \(h_I\in Im (\Psi )\) if and only if for every positive squarefree integer Q there is an automorphism in
$$\begin{aligned} Gal \big (K\big (\zeta _{Qh},W_1^{1/Qh_1}, \ldots , W_n^{1/Qh_n}\big )/K\big (\zeta _{h},W_1^{1/h_1}, \ldots , W_n^{1/h_n}\big )\big ) \end{aligned}$$which for any choice of \(i\in I\) and \(q\mid Q\) prime is not the identity on \(K(\zeta _{q h_i}, W_i^{1/q h_i})\).

(ii)
Let \(e_I\in {\mathbb {Z}}_{\geqslant 0}^n\) and \(e:=\max (e_1,\ldots , e_n)\). We have \(e_I\in Im (\Psi _\ell )\) if and only if there is an automorphism in
$$\begin{aligned} Gal \big (K\big (\zeta _{\ell ^{e+1}},W_1^{1/\ell ^{e_1+1}}, \ldots , W_n^{1/\ell ^{e_n+1}}\big )/K\big (\zeta _{\ell ^{e}},W_1^{1/\ell ^{e_1}}, \ldots , W_n^{1/\ell ^{e_n}}\big )\big ) \end{aligned}$$which for any choice of \(i\in I\) is not the identity on \(K(\zeta _{\ell ^{e_i+1}}, W_i^{1/\ell ^{e_i+1}})\).

(iii)
Let \(T\geqslant 2\) be a squarefree integer and let \(\ell \) denote its prime factors. Let \(E:=(e_{i,\ell })\in Mat ({\mathbb {Z}}_{\geqslant 0}),\) set \(H_i:=\prod _\ell \ell ^{e_{i,\ell }}\) and \(H:=lcm (H_1,\ldots , H_n)\). Then we have \(E\in Im (\Psi _T)\) if and only if there is an automorphism in
$$\begin{aligned} Gal \big (K\big (\zeta _{HT},W_1^{1/H_1T}, \ldots , W_n^{1/H_nT}\big )/K\big (\zeta _{H},W_1^{1/H_1}, \ldots , W_n^{1/H_n}\big )\big ) \end{aligned}$$which for any choice of \(i\in I\) and \(\ell \mid T\) prime is not the identity on \(K(\zeta _{\ell H_i}, W_i^{1/\ell H_i})\).
Proof
For (i) we can apply Theorem 4.1 with \(F = K\) because \(C_Q \ne \emptyset \) is equivalent to the existence of a suitable automorphism. We now prove (ii), the proof of (iii) being analogous. Set \(L: = K(\zeta _{\ell ^{e+1}}, W_1^{1/\ell ^{e_1 + 1}}, \ldots , W_n^{1/\ell ^{e_n + 1}})\). If a suitable automorphism \(\sigma \) exists, then by the Chebotarev density theorem there are infinitely many primes \({\mathfrak {p}}\) of K (in the domain of \(\Psi \) and not ramifying in L) such that \(\sigma \in \left( \frac{L/K}{{\mathfrak {p}}}\right) \) and hence \(\Psi _{\ell }({\mathfrak {p}}) = e_I\). Else, for all \({\mathfrak {p}}\) in the domain of \(\Psi _{\ell }\) not ramifying in L/K and for all \(\sigma \in \left( \frac{L/K}{{\mathfrak {p}}}\right) \), the automorphism \(\sigma \) is not as in the statement: if it does not fix \(\zeta _{\ell ^e}\) and \(W_i^{1/\ell ^{e_i}}\), then the index of \(W_i \bmod {\mathfrak {p}}\) is not divisible by \(\ell ^{e_i}\); else it fixes some of \(K(\zeta _{\ell ^{e_i + 1}}, W_i^{1/\ell ^{e_i + 1}})\) hence the index of \(W_i \bmod {\mathfrak {p}}\) is divisible by \(\ell ^{e_i+1}\). We conclude because by Theorem 4.1 (see Remark 4.2) the nonempty preimages of \(\Psi _\ell \) are sets of positive density. We may also conclude without assuming GRH because the primes \({\mathfrak {p}}\) ramifying in \(K(\zeta _{\ell ^\infty }, W^{1/\ell ^\infty })/K\) are excluded from the domain of \(\Psi _{\ell }\) [if \(v_{{\mathfrak {p}}}(w)\ne 0\) for some \(w\in W\), this is because \(W \bmod \mathfrak {p}\) is not welldefined; if \({\mathfrak {p}}\) is over \(\ell \) and \(\zeta _\ell \in K\), because \(\ell \) ramifies in K] or \({\mathfrak {p}}\) is over \(\ell \) and \(\zeta _\ell \notin K\), which means that \(\Psi _{\ell }({\mathfrak {p}})\) is the constant tuple 0, and the preimage of this tuple is infinite. \(\square \)
Remark 4.4
In the Index Map Problem we assume that the groups have positive rank. If some of them is finite, then the preimage of any finite subset of \({\mathbb {Z}}^n_{>0}\) is finite, see Remark 6.7. However, the preimage of an infinite set could be an infinite set of density 0 or without a Dirichlet density. For example, consider \(K={\mathbb {Q}}\) and \(W_1=\langle 1 \rangle \), \(W_2=\langle 2 \rangle \) and the preimage of \(S\times {\mathbb {Z}}_{>0}\), where \(S\subset {\mathbb {Z}}_{>0}\) is any infinite set such that \(\{s+1 \mid s\in S\}\) is a set of primes of density 0 (respectively, without a Dirichlet density).
5 Computability of the image of the index map
For \(J\subset I\) and \(i\in I\setminus J\), we call \(k_{J, i}\) the smallest positive integer z such that \(W_i^z \subset \prod _{j \in J} W_j\) (setting \(k_{J, i}=\infty \) if there is no z) and we call \(k:=lcm (k_{J,i})\) (treating \(\infty \) as 1), where the least common multiple is over \(J \subset I\) and \(i \in I \setminus J\).
Theorem 5.1
Assume GRH. Let \(h_I\in {\mathbb {Z}}_{>0}^n\), \(h:=lcm (h_1,\ldots , h_n),\) and let \(\omega \) be the number of distinct prime divisors of h. Assuming that arithmetic operations and computing prime factors can be performed in negligible time, checking whether \(h_I\in Im (\Psi )\) can be done in \(O(\omega )\) steps (the implied constant depends on \(K, F, n, W_1,\ldots , W_n)\).
Proof
Step 1. Reformulating the problem in terms of existence of a suitable automorphism, and reducing the problem to considering the Galois group corresponding to a single prime q. By Theorem 4.1 and Proposition 3.1(iii), calling Q the squarefree part of c, we have to check whether there is
such that \(\sigma \mid _F\,\in C\) and \(\sigma \) is the identity on \(K(\zeta _{h_i}, W_i^{1/h_i})\) but for each choice of \(i\in I\) and \(q\mid Qh\) prime it is not the identity on \(K(\zeta _{qh_i}, W_i^{1/qh_i})\). Notice that, if \(F'/F\) is a finite Galois extension and \(C'\subset Gal (F'/K)\) consists of the automorphisms whose restriction to F is in C, then we do not change our problem by replacing F by \(F'\) and C by \(C'\). Thus, up to extending F independently of h (and replacing C), by Proposition 3.1(ii) the cyclotomicKummer extensions for W made with coprime parameters are linearly disjoint over F. Hence we may consider the problem separately for different values of \(q \mid Qh\).
Step 2. Reformulating the existence of a suitable automorphism in terms of generators of the groups \(W_i\). Call \(e_I:=v_q(h_I)\). By symmetry we may assume \(e_I\) is nonincreasing. The splitting conditions then become: for all \(i\in I\),
We exclude the case where for some \(j > i\) we have \(W_j^{k}\subset \langle W_1,\ldots , W_{i}\rangle \) and \(e_j <e_{i}v_q(k\tau )\), as in this case we have \(h_I \not \in Im \Psi \): if \(\sigma \) is the identity on \(K(\zeta _{q^{e_{i'}}}, W_i^{1/q^{e_{i'}}})\) for all \(i' \leqslant i\), then necessarily \(\sigma \) is also the identity on \(K(\zeta _{q^{e_j + 1}}, W_j^{1/q^{e_j + 1}})\).
We partition I into intervals \(T_j\), their starting points being 1 and those \(i\in I\) such that \(e_i <e_{i1}v_q(k\tau )\). We call \(\rho _i:=v_q(\# W_{i, tors})\), \({\mathcal {W}}_j:=\langle W_i : i\in T_j\rangle \), and for \(i\in T_j\) we let \(S_i\subset W_i\) be a minimal set (it is nonempty) of multiplicatively independent elements such that \(W_i\subset \langle K^\times _{{{\,\textrm{tors}\,}}}, S_i, {\mathcal {W}}_1^{1/k}, \ldots , {\mathcal {W}}_{j1}^{1/k}\rangle \).
We prove that (3) holds for all \(i \in I\) if and only if for all \(i \in I\) we have
Notice that, for every \(x\geqslant 1\), fixing \({\mathbb {Q}}(\zeta _{q^x}, W_i^{1/q^{x}})\) implies fixing \(S_i^{1/q^{x}}\) and \(\zeta _{q^{x + \rho _i}}\). Moreover, since \(W_i \subset \langle K_{{{\,\textrm{tors}\,}}}, S_i, {\mathcal {W}}_1^{1/k}, ... , {\mathcal {W}}_{j1}^{1/k} \rangle \), then \(W_i^{1/q^{y}}\) is fixed if \(\zeta _{q^{y + \rho _i}}\), \(S_i^{1/q^{y}}\), \(\zeta _{q^{y+v_q(k\tau )1}}\) and \({\mathcal {W}}_{j'}^{1/q^{y+v_q(k\tau )1}}\) for \(j' < j\) are all fixed. Taking \(x = e_i\) and \(y = e_i + 1\), the ‘only if’ implication follows. The converse is proven similarly (by induction on i).
Step 3. Reformulating the existence of a suitable automorphism in terms of systems of linear (in)congruences. Fixing j, let \(e := e_{\min T_j}\), and let \(\alpha _1, \ldots , \alpha _r\in {\mathcal {W}}_j\) be multiplicatively independent and such that \(S_i\subset \langle K^\times _{{{\,\textrm{tors}\,}}},\alpha _1, \ldots , \alpha _r\rangle \) for all \(i\in T_j\). For any \(\sigma \) define \({\textbf{x}}:=(x_0,x_1,\ldots , x_r)\) as the smallest positive integers such that
For \(i\in T_j\) and \(w_{i,i'}\in S_i\) define \({\textbf{f}}_{i, i'} = (f_0, \ldots , f_r)\) such that
The splitting conditions concerning some \(i\in T_j\) amount to
Here \(\langle \cdot , \cdot \rangle \) is the usual inner product of vectors.
Step 4. Showing how to efficiently decide whether the congruence problem has a solution.
Considering all \(i\in T_j\) we have various systems of linear incongruences, and the question is if at least one system has a solution in common with the linear congruences. Since there are only boundedly many system of incongruences, we may consider them separately and thus fix one of them.
Note that different sets \(T_j\) give different systems of (in)congruences. The only common variable between these systems is \(x_0\). Any common solution to the systems necessarily has \(v_q(x_0  1) \geqslant \max _{i}(e_i + \rho _i)\), and thus we cannot use the incongruence for \(x_0\) if \(e_i + \rho _i\) is smaller than its maximal value over \(i\in I\). Hence we essentially have no common variables between the different systems, and thus we may now fix some \(T_j\).
Let \(q^N\) and S be the integer and the set from Lemma 3.3, and consider separately each \({\textbf{s}}:=(s_0, \ldots , s_r)\in S\). Setting \(x_i := q^Ny_i + s_i\) with \({\textbf{y}}:=(y_0,y_1,\ldots , y_r)\), and rewriting \(\langle {\textbf{x}}, {\textbf{f}}_{i, i'} \rangle \equiv 0\) as \(\langle {\textbf{y}}, q^N{\textbf{f}}_{i, i'}\rangle \equiv \langle {\textbf{s}}, {\textbf{f}}_{i, i'}\rangle \), we get a system of the form
for some \(J'\subset J\subset T_j\), for some vectors \({\textbf{v}}_i\), and for some integers \(c({\textbf{v}}_i)\) and \(e({\textbf{v}}_i)\in \{e_i, e_i+\rho _i\}\). The vectors \({\textbf{v}}_i\) consist of the vectors \(q^N{\textbf{f}}_{i, i'}\) and of the vector \((q^N, 0, \ldots , 0)\) to express the conditions for \(x_0\). Notice that \(\vert e({\textbf{v}}_i)  e({\textbf{v}}_{i'}) \vert \leqslant n v_q(k\tau )\) for all \(i,i'\in J\).
Let \({\mathcal {J}}\subset J\) be a maximal subset such that \({\textbf{v}}_i, i\in {\mathcal {J}}\) are \({\mathbb {Q}}\)linearly independent, and among the possible choices for \({\mathcal {J}}\), take any which minmizes \(\sum _{i \in {\mathcal {J}}} e({\textbf{v}}_i)\). For \(i\in J\) let \({\textbf{u}}_i\) be the vector of coefficients to express \({\textbf{v}}_i\) as a \({\mathbb {Q}}\)linear combination of \({\textbf{v}}_j, j \in {\mathcal {J}}\).
We claim that as \({\textbf{y}}\) varies over integer vectors, the values for \({\textbf{z}} := (\langle {\textbf{y}}, {\textbf{v}}_i \rangle )_{i \in {\mathcal {J}}}\) are all preimages of the (nonempty) set \({\mathcal {M}}\) of their reductions modulo M, for some positive integer M. Furthermore, M and \({\mathcal {M}}\) can be calculated with an explicit finite procedure. Indeed, consider an integer tuple \((n_i)_{i \in {\mathcal {J}}}\) and the system
One can solve the system over \({\textbf{y}} \in {\mathbb {Q}}^{{\mathcal {J}}}\), writing the components of \({\textbf{y}}\) as \({\mathbb {Q}}\)linear combinations of \(n_i\) (the coefficients depending on \({\textbf{v}}_i\)). Letting M be the least common denominator of these coefficients, the components of \({\textbf{y}}\) are integers if and only if certain linear combinations of \(n_i\) are divisible by M. This holds if and only if \((n_i)_{i \in {\mathcal {J}}}\) modulo M lies in a certain set \({\mathcal {M}} \subset ({\mathbb {Z}}/M{\mathbb {Z}})^{{\mathcal {J}}}\), which may be computed by letting \({\textbf{y}}\) range through \(\{0, 1, \ldots , M1\}^{{\mathcal {J}}}\) and considering the values of \((\langle {\textbf{y}}, {\textbf{v}}_i\rangle )_{i \in {\mathcal {J}}}\) modulo M.
Since \(\langle {\textbf{y}}, {\textbf{v}}_i \rangle = \langle {\textbf{z}}, {\textbf{u}}_i \rangle \) we may rewrite (5) as
By the Chinese remainder theorem we replace the last condition by \(({\textbf{z}} \bmod {q^{v_q(M)}}) \in ({\mathcal {M}} \bmod {q^{v_q(M)}})\) without changing the solvability of the system. As \({\textbf{u}}_i, i \in {\mathcal {J}}\) are the standard basis for \({\mathbb {Q}}^{{\mathcal {J}}}\), \({\textbf{z}}\) is determined modulo \(q^{\min _{i \in {\mathcal {J}}} e({\textbf{v}}_i)}\).
Suppose first that q is large enough [namely \(q > J'\), \(N = v_q(\tau ) = v_q(M) = 0\), and the entries of \({\textbf{u}}_i, i\in J\) are 0 or have qadic valuation 0]. Then (6) becomes
As \({\mathcal {J}}\) is chosen such that \(\sum _{i \in {\mathcal {J}}} e({\textbf{v}}_i)\) is minimized (among all \({\mathcal {J}}\) of maximal size), all congruences follow from the necessary conditions \(z_i \equiv 0 \pmod {q^{e({\textbf{v}}_i)}}\) for all \(i \in {\mathcal {J}}\). (Indeed, otherwise for some \(i \in J\) and \(i' \in {\mathcal {J}}\) we have \({\textbf{u}}_{i, i'} \ne 0\) and \(e({\textbf{v}}_i) < e({\textbf{v}}_{i'})\), and one could make \(\sum _{i \in {\mathcal {J}}} e({\textbf{v}}_i)\) smaller by replacing \({\mathcal {J}}\) with \({\mathcal {J}} \cup \{i\} \setminus \{i'\}\).) As for the incongruences, there are \(q^{{\mathcal {J}}}\) values of \({\textbf{z}}\) where \(0 \leqslant z_i < q^{e({\textbf{v}}_i) + 1}\) and \(z_i \equiv 0 \pmod {q^{e({\textbf{v}}_i)}}\) for all \(i\in {\mathcal {J}}\). If for some \(i\in J'\) we have \(\min _{j : {\textbf{u}}_{i, j} \ne 0} e({\textbf{v}}_j) \geqslant e({\textbf{v}}_i) + 1\), then the incongruence for that index i is not solvable, else the system is solvable, as each incongruence excludes at most \(q^{{\mathcal {J}}1}\) values. All large values of q may thus be solved at once (the \({\textbf{u}}_i\)’s depend on q only through the ordering of the \(e({\textbf{v}}_i)\)’s).
For each of the finitely many remaining values of q we may check the solvability of (6) by brute force. Indeed: Letting V denote \(\max _{i \in J} \max _{j \in {\mathcal {J}}} v_q({\textbf{u}}_{i, j})\), if \(v_q(M) > \min _{i \in J} e({\textbf{v}}_i)  V\), this is because the moduli are bounded in terms of \(K, F, n, W_1,\ldots , W_n\); else, we have \(v_q(M) \leqslant \min _{i \in J} e({\textbf{v}}_i)  V\), and the congruences determine \((z_i \bmod q^{\max (e({\textbf{v}}_i)  V, 0)})\). Hence the condition on \(({\textbf{z}} \bmod {q^{v_q(M)}})\) is either trivially satisfied or impossible, moreover the number of possible values for \((z_i \bmod q^{\max e({\textbf{v}}_i) + V + 1})\) to be checked is at most \(q^{\max e({\textbf{v}}_i) + V + 1  \min e({\textbf{v}}_i)}\). \(\square \)
Remark 5.2
Assume GRH. Let \(e_I\in {\mathbb {Z}}_{\geqslant 0}^n\) and consider the question whether \(Gal ({\bar{K}}/K)\) contains an automorphism whose restriction to F is in C and that for every \(i\in I\) it is the identity on \(K(\zeta _{q^{e_i}}, W_i^{1/q^{e_i}})\) but not on \(K(\zeta _{q^{e_i}}, W_i^{1/q^{e_i+1}})\).

(i)
By the proof of Theorem 5.1, the answer does not depend on q for all q larger than a constant (depending on \(K, F, n, W_1, \ldots , W_n\) and computable with an explicit finite procedure).

(ii)
By the proof of Theorem 5.1, there is a constant c (depending on K, F, n, \(W_1, \ldots , W_n\) and computable with an explicit finite procedure) such that for any q the answer is not affected by changing \(e_I\) as follows: for some nonempty \(J \subset I\) such that
$$\begin{aligned} d_J := \min _{j \in J} e_j  \left( c + \max _{i \in I \setminus J} e_i\right) > 0 \end{aligned}$$we replace \(e_i\) by \(e_id_J\) for all \(i\in J\). Indeed, if \(c > v_q(k \tau )\), then the choice of the intervals \(T_j\) in the proof stays the same, and if further \(c > v_q(M) + V\) for any M and V as in the proof, we are either in the case of “large” q or the case \(v_q(M) \leqslant \min _{i \in J} e({\textbf{v}}_i)  V\) hence whether the system is solvable depends on the differences \(e({\textbf{v}}_i)  e({\textbf{v}}_j)\) but not on the specific values of \(e({\textbf{v}}_i)\).

(iii)
If \(F=K\), then we can answer by applying the Chebotarev density theorem (and the inclusion–exclusion principle), computing the degrees of the given cyclotomicKummer extensions and of their various compositum fields.
Example 5.3
Assume GRH. Consider \(K = F = {\mathbb {Q}}(i)\), \(C = \{\text {id}_K\}\), \(\alpha =2 + i\), \(\beta = 3+2i\), and
By Example 7.3 for every \(m\geqslant 1\) we have \([K(\zeta _{\infty }, \alpha ^{1/m}, \beta ^{1/m}):K(\zeta _{\infty })]=m^2\). Hence for \(h_I \in {\mathbb {Z}}_{> 0}^4\) we have \(h_I \in Im (\Psi )\) if and only if \(v_q(h_I) \in Im (\Psi _q)\) for every q. Indeed, by Theorem 4.1 the former condition means \(C_Q \ne \emptyset \) for all squarefree Q, and this in turn means (by the maximality of the above Kummer extensions) that \(C_q \ne \emptyset \) for all primes q, i.e. \(v_q(h_I) \in Im (\Psi _q)\) for all q.
For \(q\ne 2\), \(Im (\Psi _q)\) consists of those \(e_I\in {\mathbb {Z}}_{\geqslant 0}^4\) such that \(e_1 = e_2 = e_3 = e_4\) or there is a permutation f such that \(e_{f(1)} > e_{f(2)} = e_{f(3)}= e_{f(4)}\). We sketch the proof of this fact, supposing for simplicity that \(e_I\) is nonincreasing. Set \(e:=e_1\). We use Corollary 4.3 to reduce the question of whether \(e_I \in Im (\Psi _q)\) to finding an automorphism \(\sigma \) of \(L := K(\zeta _{q^{e + 1}}, W^{1/q^{e + 1}})/K\) that for every i is the identity on \(K(\zeta _{q^{e_i}}, W_i^{1/q^{e_i}})\) but not on \(K(\zeta _{q^{e_i+1}}, W_i^{1/q^{e_i+1}})\). Since we have \([L : K] = \varphi (q^{e+1})q^{2(e+1)}\), for any \({\textbf{x}}=(x_0, x_1, x_2)\) such that \(q\not \mid x_0\) there exists an automorphism \(\sigma \) of L such that
We now look for \({\textbf{x}}\) satisfying the congruences
and some incongruences, e.g. for \(W_4\) we can have \(x_0 \not \equiv 1 \pmod {q^{e_4 + 1}}\) or \(2x_1 + x_1 \not \equiv 0 \pmod {q^{e_4 + 1}}\). Partitioning I into intervals regrouping the indices i with the same \(e_i\), we get either \(\{1\}, \{2,3,4\}\) or \(\{1,2,3,4\}\), else there is no \({\textbf{x}}\) as requested. In the former case there are solutions, as we require \(x_0 \equiv 1 \pmod {q^{e_1}}, x_1 \equiv 0 \pmod {q^{e_1}}, x_2 \equiv 0 \pmod {q^{e_2}}\) and \(x_2 \not \equiv 0 \pmod {q^{e_2+1}}\), and \(x_0 \not \equiv 1 \pmod {q^{e_1+1}}\) or \(x_1 \not \equiv 0 \pmod {q^{e_1+1}}\). In the latter case there are solutions, as we require \(x_0 \equiv 1 \pmod {q^{e_1}}, x_1 \equiv x_2 \equiv 0 \pmod {q^{e_1}}\), and \(x_0 \not \equiv 1 \pmod {q^{e_1+1}}\) (which can be satisfied for any \(q \geqslant 3\)) or \(x_1, x_2, x_1+x_2, 2x_1 + x_2 \not \equiv 0 \pmod {q^{e_1+1}}\) (which can be satisfied for \(q > 3\)).
Theorem 5.4
Assume GRH. The image of the index map is computable with an explicit finite procedure. Moreover, there exists a positive squarefree integer Q (computable with an explicit finite procedure) such that the following holds:

(i)
The image of \(\Psi _\ell \) for \(\ell \not \mid Q\) does not depend on \(\ell \).

(ii)
For \(h_I\in {\mathbb {Z}}_{>0}^n,\) we have \(h_I\in Im \Psi \) if and only if \(v_Q(h_I)\in Im \Psi _Q\) and \(v_\ell (h_I)\in Im \Psi _\ell \) for all \(\ell \not \mid Q\).
Proof
Assertion (i) is a consequence of Remark 5.2(i) (and mimicking the proof of Theorem 5.1 directly in the \(\ell \)adic setting is likely to provide an alternative proof without assuming GRH). Indeed, as in the proof of Theorem 5.1, we can set Q to be the squarefree part of the constant from Proposition 3.1(iii), where we require the additional condition with F (enlarging F as in the proof of Theorem 5.1 and replacing C accordingly).
We now prove (ii). Corollary 4.3(i) allows to reduce the condition \(h_I\in Im \Psi \) to the existence of a certain automorphism. By the independence of Kummertype extensions (see Proposition 3.1) we may reduce the problem to extensions w.r.t. \(\ell \mid Q\) and other values of \(\ell \) separately. Then we conclude by Corollary 4.3(ii)–(iii).
For the computability of the image, we are left to show that we can determine \(Im \Psi _Q\) and \(Im \Psi _\ell \) for some \(\ell \not \mid Q\). Notice that checking whether a given value is in \(Im \Psi _Q\) (respectively, \(Im \Psi _\ell \)) can be done explicitly by Corollary 4.3. We conclude because Remark 5.2(ii) reduces computing \(\Psi _\ell \) to checking a computable finite list of values: to determine \(Im \Psi _Q\), notice that in the proof of Theorem 5.1 we extend F so that the problems with different \(\ell \) are independent (absorbing the interactions into the condition on the Artin symbol), thus primes \(\ell \mid Q\) can be checked individually. \(\square \)
6 The index map for separated groups
Remark 6.1
Assume GRH, and recall Corollary 4.3. If the \(W_i\)’s have separated Kummer extensions w.r.t. \(x_I\in {\mathbb {Z}}_{>0}^n\), then for all \(y_I\in {\mathbb {Z}}_{>0}^n\) such that \(x_I\in \otimes y_I\) we have \(y_I\in Im (\Psi )\) if and only if there is an automorphism in
which, for every prime q and \(i\in I\) such that \(q y_i\mid x_i\), is not the identity on \(K(\zeta _{q y_i}, W_i^{1/q y_i})\). Indeed, the difference w.r.t. Corollary 4.3(i) is considering the last condition only for certain pairs (q, i). This will be sufficient because those pairs (q, i) such that \(q y_i\mid x_i\) are taken care of by assumption, and for the remaining ones we may perform an extension argument: in case \(q y_i\not \mid x_i\), we can consider the Kummer extension with parameter \(qx_i\) in place of \(x_i\) and by separatedness we can extend the given automorphism so that the requested condition also holds for this pair (q, i). Similarly, if the \(W_i\)’s have \(\ell \)separated Kummer extensions w.r.t. \(x_I\in {\mathbb {Z}}_{\geqslant 0}^n\) and \(y_I\in \mathbb Z_{\geqslant 0}^n\) is such that \(x_I\in \oplus y_I\), then we have \(y_I\in Im (\Psi _\ell )\) if and only if there is an automorphism in
which, for every \(i\in I\) such that \(y_i< x_i\), is not the identity on \(K(\zeta _{\ell ^{y_i+1}}, W_i^{1/\ell ^{y_i+1}})\).
Theorem 6.2
Assume GRH. The following are equivalent for \(x_I\in \mathbb Z_{>0}^n\) (respectively, for \(x_I\in {\mathbb {Z}}_{\geqslant 0}^n)\):

(i)
The \(W_i\)’s have separated (respectively, \(\ell \)separated) Kummer extensions w.r.t. \(x_I\).

(ii)
We have \(\otimes x_I \subset Im (\Psi )\) (respectively, \(\oplus x_I \subset Im (\Psi _\ell ))\).

(iii)
For each \(y_I \in {\mathbb {Z}}_{>0}^n\) (respectively, \({\mathbb {Z}}_{\geqslant 0}^n)\) the following holds:
$$\begin{aligned} y_I \in Im (\Psi ) \Leftrightarrow \gcd (y_I, x_I)\in Im (\Psi ) \end{aligned}$$(respectively, \(y_I\in Im (\Psi _\ell ) \Leftrightarrow \min (y_I, x_I)\in Im (\Psi _\ell ))\).
Proof
For any \(X_I\in {\mathbb {Z}}_{>0}^n\) such that the Kummer extensions are separated w.r.t. \(X_I\) we have \(X_I\in Im (\Psi )\), see the above remark. Thus (i) implies (ii) because \(\otimes x_I \subset Sep\) by Proposition 3.2. Reasoning similarly to the previous remark, (i) implies (iii). For every \(x_I\) we have \(Im (\Psi )\cap \otimes x_I\ne \emptyset \) (consider primes of K that split completely in \(K(\zeta _{lcm (x_i)}, W^{1/lcm (x_i)})\)) and hence (iii) implies (ii). If (i) does not hold, then w.l.o.g. there are positive integers \(N_1,N\) and a prime q such that \(x_1\mid N_1\), \(lcm (x_1,\ldots , x_n, q N_1)\mid N\) and
hence \((N_1, N,\ldots , N)\notin Im (\Psi )\) and (ii) does not hold. The equivalence of the respective assertions is analogous. \(\square \)
Remark 6.3
To determine the image of \(\Psi \) (respectively, \(\Psi _\ell \)) for separated groups, it suffices to compute the image of \({\mathfrak {p}}\) for the finitely many excluded primes (see Sect. 2) and, by Proposition 3.2(iii) and Theorem 6.2, apply finitely many times Remark 6.1. Moreover, \(Im (\Psi )\) is the preimage of its reduction modulo M for some positive integer M if and only if the groups are separated, see (iii) and (ii) for \(\ell \not \mid M\) of Theorem 6.2.
We then determine the image of the index map \(p \mapsto Ind _p(a)\) over \({\mathbb {Q}}\). Notice that we can write \(a\in {\mathbb {Q}}^\times \setminus \{\pm 1\}\) uniquely in the form
where \(\epsilon , \delta \in \{0,1\}\), \(b \in {\mathbb {Q}}^{\times }\), \(d\geqslant 0\), with at least one of \(\delta = 1\) and \(T \ne 1\) holding, and where T is a squarefree integer of the form
where \(p_i, q_j\) are prime numbers.
In the following result we see sets of excluded values for the index map: \(E_{\square }\) is because the index for squares cannot be odd; \(E_{T}\) is because in the given case \(T\mid Ind _p(a)\) implies \(2\mid Ind _p(a)\); \(E_{2T}\) is because in the given case \(2T\mid Ind _p(a)\) implies \(4\mid Ind _p(a)\); \(E_{3,T}\) is because in the given case \(2^m(T/3) \mid Ind _p(a)\) implies \(3 \mid Ind _p(a)\).
Theorem 6.4
Let \(a \in {\mathbb {Q}}^{\times } \setminus \{\pm 1\}\). The image of the index map \(p \mapsto Ind _p(a)\) as p ranges over the odd primes with \(v_p(a) = 0\) is
where, writing a as in (8), we define
(the sets have to be considered empty if the conditions are not met) and
Notice that the image does not change by excluding a set of primes of density 0 from the given index map.
Remark 6.5
After this work was accepted to publication, the authors noticed that Theorem 6.4 has previously been given by Lenstra [9, (8.9) to (8.13)] (though a detailed proof has not been provided there).
Proof
Let D be the largest odd integer such that \(a\in (\mathbb Q^\times )^D\). We set \(E:=\max (3\delta , d+2)\) and \(Z:=2^{E}lcm (T,D)\). The group \(\langle a \rangle \) has separated Kummer extensions w.r.t. Z: following [16, 17] we have to consider the \(\ell \)adic failure [we have \(\alpha ^{1/{\ell ^{x+1}}}\notin {\mathbb {Q}}(\zeta _{\ell ^{\infty }}, \alpha ^{1/{\ell ^{x}}})\) for \(\ell \ne 2\), \(x\geqslant {v_\ell (D)}\) and for \(\ell =2\), \(x\geqslant E\)] and the 2adelic failure [the related cyclotomic field is \({\mathbb {Q}}(\zeta _{2^ET})\)].
By Theorem 6.2 we only have to determine the positive divisors z of Z that are values for the index map, namely such that \(Gal ({\mathbb {Q}}(\zeta _{Z}, a^{1/Z})/{\mathbb {Q}}(\zeta _{z}, a^{1/z}))\) contains an automorphism which for every prime divisor q of Z/z is not the identity on \({\mathbb {Q}}(\zeta _{qz}, a^{1/qz})\). Fix a positive divisor z of Z and call \(v:=v_2(z)\) and \(V:=v_3(z)\). For a prime \(q>3\) dividing Z/z we have \(\zeta _{zq}\notin {\mathbb {Q}}(\zeta _{Z/q}, a^{q/Z})\) so we are left to check whether \(Gal ({\mathbb {Q}}(\zeta _{6z}, a^{1/6z})/{\mathbb {Q}}(\zeta _{z}, a^{1/z}))\) contains an automorphism which is not the identity on \({\mathbb {Q}}(\zeta _{2z}, a^{1/2z})\) if \(v<E\) and it is not the identity on \({\mathbb {Q}}(\zeta _{3z}, a^{1/3z})\) if \(V< v_3(lcm (T,D))\). Such an automorphism exists unless \(v<E\) and \({\mathbb {Q}}(\zeta _{2z}, a^{1/2z})={\mathbb {Q}}(\zeta _{z}, a^{1/z})\), or \(V<v_3(lcm (T,D))\) and \({\mathbb {Q}}(\zeta _{3z}, a^{1/3z})={\mathbb {Q}}(\zeta _{z}, a^{1/z})\).
We first determine when \({\mathbb {Q}}(\zeta _{2z}, a^{1/2z}) = {\mathbb {Q}}(\zeta _z, a^{1/z})\) holds. We will show that this is the case precisely when we are in one of the cases indicated by \(E_{\square }, E_{T}\) and \(E_{2T}\). We necessarily have \(v \leqslant 2\) because \(\zeta _{2^{v+1}} \not \in {\mathbb {Q}}(\zeta _{2^vw})\) for w odd and \({\mathbb {Q}}(\zeta _z, a^{1/z})\cap {\mathbb {Q}}(\zeta _{2^\infty })\) is contained in \({\mathbb {Q}}(\zeta _{2^{\max (v+1,3)}})\).
If \(v = 0\), the condition becomes \(\sqrt{a} \in {\mathbb {Q}}(\zeta _z)\): this holds when either \(d \geqslant 1\) and \(\epsilon = 0\), or \(d = \delta = \epsilon = 0\) and \(T \mid z\). These obstructions correspond to the exceptional sets \(E_{\square }\) and \(E_{T}\).
If \(v = 1\), the condition becomes \({\mathbb {Q}}(\zeta _{4}, a^{1/4}) \subset {\mathbb {Q}}(\zeta _{w}, a^{1/2})\) where \(w:= z/2\) is odd. The former field must be abelian over \({\mathbb {Q}}\) hence \(d \geqslant 1\). So the latter field contains \(\zeta _4\) only if \(\epsilon = 1\). Finally, from \(a^{1/4} \in {\mathbb {Q}}(\zeta _{4w})\) we obtain \(d = 1\), \(\delta = 1\), \(T \mid z\). This results in the set \(E_{2T}\).
If \(v = 2\), the condition becomes \({\mathbb {Q}}(\zeta _{8}, a^{1/8}) \subset {\mathbb {Q}}(\zeta _{4w}, a^{1/4})\) where \(w:= z/4\) is odd. If \(\epsilon = 0\), to get \(\zeta _8\) we need \(d \leqslant 1\), and to get \(a^{1/8}\) we need \(d \geqslant 2\), a contradiction. If \(\epsilon = 1\) we similarly have \(d \geqslant 2\) and hence \(\zeta _{16} \in {\mathbb {Q}}(\zeta _{8w}, a^{1/8})\), contradicting \(\zeta _{16} \not \in {\mathbb {Q}}(\zeta _{4w}, a^{1/4})\). Hence, no obstructions arise from \(v = 2\).
We now determine when \({\mathbb {Q}}(\zeta _{3z}, a^{1/3z}) = {\mathbb {Q}}(\zeta _{z}, a^{1/z})\). This will result in the exceptional set \(E_{3,T}\). By considering the largest abelian subextensions, we have \(V = 0\), a is a cube, and \(\zeta _3 \in {\mathbb {Q}}(\zeta _z, a^{1/2^v})\). The last condition holds if and only if \(v \geqslant d+1\), \(3\mid T\), \((T/3) \mid z\) and \(\zeta _3 \in L(a^{1/2^{d+1}})\), where L is the maximal subextension of \({\mathbb {Q}}(\zeta _z)\) of exponent 2. So we must have \(L(\zeta _3) = L(a^{1/2^{d+1}}) = L(\zeta _{2^{d+2}}^{\epsilon }\sqrt{(3) \cdot 2^{\delta }})\).
For \(\epsilon = \delta = 0\), no additional conditions are necessary. For \(\epsilon = 0, \delta = 1\), we need \(v \geqslant 3\) to ensure \(L(\sqrt{3}) = L(\sqrt{6})\). For \(\epsilon = 1, \delta = 0\) we similarly need \(v \geqslant d+2\).
Finally, for \(\epsilon = \delta = 1\) we need for \(d = 1\) (respectively, \(d\ne 1\)) that \(v \geqslant 2\) [respectively, \(v \geqslant \max (d+2, 3)\)]. Indeed, \(L(\sqrt{3}) = L(\zeta _{2^{d+2}}\sqrt{6})\) holds if and only if \(\zeta _{2^{d+2}}\sqrt{2} \in L\). We conclude because the smallest cyclotomic extension in which \(\zeta _{2^{d+2}}\sqrt{2}\) lies is: \({\mathbb {Q}}(\zeta _8)\) for \(d = 0\); \({\mathbb {Q}}(\zeta _4)\) for \(d = 1\); \({\mathbb {Q}}(\zeta _{2^{d+2}})\) for \(d \geqslant 2\). \(\square \)
The strategy of the above proof should extend to a number field K known very explicitly (e.g. a quadratic field), and we could replace a by a finitely generated subgroup of \(K^\times \).
Example 6.6
The index map \(p \mapsto Ind _p(2)\) is surjective by Theorem 6.4, and also by Theorem 6.2, as its image contains all positive divisors of 8 (consider \(p=3, 7, 113, 73\)).
The image of the index map \(p \mapsto Ind _p(100)\) consists of the positive integers not congruent to 10 modulo 20. These values must be excluded, as \(100\) is minus a square, thus \(10 \mid Ind _p(100)\) implies \(p \equiv 1 \pmod {20}\) and hence \(20 \mid Ind _p(100)\) because \(100=(\sqrt{5}(1+\zeta _4))^4\).
For \(a=3\) [respectively, \(a=(3)^3\)] and \(p\geqslant 5\) we have \(3\mid Ind _p(a)\) only if (respectively, if and only if) \(2\mid Ind _p(a)\) because \({\mathbb {Q}}(\zeta _2, a^{1/2})\) is contained in (respectively, equals) \({\mathbb {Q}}(\zeta _3, a^{1/3})\).
Remark 6.7
Over K, fix some positive integer \(m\mid \tau \) and consider the index map \({\mathfrak {p}} \mapsto Ind _\mathfrak {p}(\zeta _m)\) for \(\mathfrak p\not \mid mO_K\). Its image has density 0 inside \({\mathbb {Z}}_{>0}\) because \(Ind _\mathfrak {p}(\zeta _m)=(N{\mathfrak {p}}1)/m\). For \(K={\mathbb {Q}}\) we have \(Ind _2(1)=1\) and \(Ind _p(1)=(p1)/2\) for p odd, while for \(K={\mathbb {Q}}(\zeta _4)\) we have \(Ind _{1+\zeta _4}(\zeta _4)=1\) and \(Ind _{{\mathfrak {p}}}(\zeta _4)\) is \((p1)/2\) (respectively, \((p^21)/2\)) if \({\mathfrak {p}}\) lies over a prime \(p\equiv 1 \pmod 4\) respectively, \(p\equiv 3 \pmod 4\)). These examples show that the image of the index map for one or several finite groups is easy to describe, however the description is not explicit. Thus, if we remove from the Index Map Problem the assumption that the groups have positive rank, then we may not be able to compute the image of the index map with a finite procedure.
7 Examples of groups with surjective index map
Proposition 7.1
If \(K \ne {\mathbb {Q}},\) then there exists a sequence \((\alpha _i)_{i \in {\mathbb {Z}}_{> 0}}\) with \(\alpha _i \in K^{\times }\) for all \(i > 0\) which are algebraic integers and not units, whose norms \(N(\alpha _i)\) are pairwise coprime, and such that for all positive integers r, n we have
Assuming GRH, for any distinct \(i_1,\ldots , i_r\) the index map \({\mathfrak {p}} \rightarrow (Ind _{{\mathfrak {p}}}(\alpha _{i_1}), \ldots , Ind _{{\mathfrak {p}}}(\alpha _{i_r}))\) is surjective onto \(\mathbb Z_{>0}^r\).
Proof
By Theorem 6.2 the second assertion follows from (9), and it suffices to show (9) when \(n=\ell \) is prime. Call z the squarefree part of \(2\Omega \). For all \(\ell \mid z\) at once we construct by induction \(\alpha _{\ell ,i}\) as in Lemma 7.2 and with pairwise coprime norms. The algebraic integers \(\alpha _i:=\prod _{\ell \mid z} \alpha _{\ell , i}^{z/\ell }\) are not units and have coprime norms. Then (9) holds for \(\ell \mid z\) because given integers \(z_j\) not all divisible by \(\ell \), no \(\ell \)th root of \(\prod _j \alpha _j^{z_j}\), equivalently of \(\prod _j \alpha _{\ell ,j}^{z_jz/\ell }\), is in \(K(\zeta _\infty )\) because there are distinct primes \(q_j\) such that there is no t as in the statement of Lemma 7.2 for \(q:=q_j\) and \(\alpha :=\alpha _{\ell , j}\). It also holds for \(\ell \not \mid z\) by Proposition 3.5, Remark 3.4, and [1, Theorem 18], as for every i there is some prime \(\mathfrak p\) of K such that \(\ell \not \mid v_{{\mathfrak {p}}}(\alpha _i)\) and \(\ell \mid v_{{\mathfrak {p}}}(\alpha _j)\) for all \(j\ne i\). \(\square \)
Lemma 7.2
Let \(K \ne {\mathbb {Q}},\) fix some prime \(\ell \) and \(x>0,\) and for each \(q \equiv 1 \pmod {\ell }\) let \(1 \leqslant e_q \leqslant v_\ell (q1)\) be arbitrary and let \(g_q\) be as in Proposition 3.5. There exists an algebraic integer \(\alpha \in K^\times \) which is not a unit such that the following holds: \(\alpha ^{1/\ell }\notin K(\zeta _\infty );\) for all primes \(\mathfrak p\) of K we have \(v_{{\mathfrak {p}}}(\alpha )<\ell ;\) the norm \(N(\alpha )\) is only divisible by primes \(q>x\) not inert in K such that \(q\not \mid \ell \Omega ,\) \(q\equiv 1 \pmod \ell ;\) for some of these q there is no \(t\in {\mathbb {Z}}\) such that \(v_{\mathfrak q}(\alpha )\equiv t v_{{\mathfrak {q}}}(g^{\ell ^{e_q}}_q) \pmod \ell \) holds for all primes \({\mathfrak {q}}\) of K over q.
Proof
There are infinitely many \(q>x\) not inert in K such that \(q\not \mid \ell \Omega \), \(q\equiv 1 \pmod \ell \). Considering the primes \({\mathfrak {q}}_j\) over q, by a counting argument there is an ideal \(I_q:=\prod _j {\mathfrak {q}}_j^{z_j}\) such that \(0\leqslant z_j<\ell \) and for no \(t\in {\mathbb {Z}}\) we have \(z_j\equiv v_{{\mathfrak {q}}_j}(g^{t\ell ^{e_q}}_q)\pmod \ell \) for all j. For some set S the ideal \(\prod _{q\in S} I_q\) is principal: calling \(\alpha \) a generator, for \(q\in S\) we have \(v_{{\mathfrak {q}}_j}(\alpha )=z_j\) hence \(\alpha ^{1/\ell }\notin K(\zeta _\infty )\) by Proposition 3.5. \(\square \)
Example 7.3
If \(K={\mathbb {Q}}(\alpha _1,\ldots , \alpha _r)\) and \(\alpha _i^2\) are distinct rational primes greater than 3, or if \(K = {\mathbb {Q}}(i)\) and \(\alpha _i\) are Gaussian primes whose norms are distinct primes (see Remark 3.4), then (9) holds for every n.
Remark 7.4
We expect (9) to hold for generic elements \(\alpha _1,\ldots , \alpha _r\in K^\times \). Firstly, we expect that the \(\alpha _i\)’s are strongly \(\ell \)independent for every prime \(\ell \) and strongly 2independent over \(K(\zeta _4)\) (see [1]) hence \([K(\zeta _{\ell ^\infty }, \alpha _1^{1/\ell ^n}, \ldots , \alpha _r^{1/\ell ^n}): K(\zeta _{\ell ^\infty })]=\ell ^{nr}\). We are left to check that for \(\ell \mid \tau \) and \(\alpha :=\prod _i \alpha _i^{z_i}\) (where \(0\leqslant z_i<\ell \) and the \(z_i\)’s are not all zero) we have \(\alpha ^{1/\ell } \notin K(\zeta _{\infty })\). If we had \(\alpha ^{1/\ell } \in K(\zeta _{\infty })\), then \(\ell \mid v_\mathfrak p(\alpha )\) for all primes \({\mathfrak {p}}\) of K over \(p\not \mid \tau \) and over \(p\not \equiv 1 \pmod \ell \) (see Proposition 3.5), and there are few admissible values for the tuple \((v_{\mathfrak q}(\alpha ))\), where \({\mathfrak {q}}\) varies over the primes of K over any fixed \(q \equiv 1 \pmod \ell \) not inert in K such that \(q\ne \ell \Omega \) (see the proof of Lemma 7.2). Generically, the nonzero valuations of the \(\alpha _i\)’s are (with few exceptions) 1 and correspond to primes of K of degree 1 lying over distinct primes q. For such q, if \(z_i\ne 0\), then the tuple \((v_{{\mathfrak {q}}}(\alpha ))\) for the primes \({\mathfrak {q}}\) over q has one nonzero entry hence \(\alpha ^{1/\ell } \notin K(\zeta _{\infty })\) by [6, Theorem 12].
Example 7.5
The index map is not surjective if \(K={\mathbb {Q}}\) and \(n\geqslant 2\). If \(a,b\in {\mathbb {Q}}^\times \), then the index map \(p\mapsto (Ind _p(a), Ind _p(b))\) is not surjective because, considering the fraction \(a=\frac{n}{d}\), the condition \(4 nd \mid Ind _p(b)\) implies \(p\equiv 1 \pmod {4nd}\) hence \(2\mid Ind _p(a)\). Similarly, if \(W_1=\left\langle \frac{n_1}{d_1}, \ldots , \frac{n_r}{d_r}\right\rangle \), then we cannot have \(2\not \mid Ind _p(W_1)\) and \(4n_1d_1\cdots n_rd_r\mid Ind _p(W_2)\).
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Acknowledgements
We would like to thank Fritz Hörmann, Flavio Perissinotto, and Pietro Sgobba for helpful discussions. We also thank the two anonymous referee for their valuable work and their insights. The first author was supported by the Emil Aaltonen foundation and worked in the Finnish Centre of Excellence in Randomness and Structures (Academy of Finland Grant No. 346307).
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Järviniemi, O., Perucca, A. Unified treatment of Artintype problems. Res. number theory 9, 10 (2023). https://doi.org/10.1007/s40993022004186
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DOI: https://doi.org/10.1007/s40993022004186
Keywords
 Artin’s primitive root conjecture
 Kummer theory
 Multiplicative order
 Galois theory
 Chebotarev density theorem