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    $\begingroup$ Just to spell it out more concretely, the fact that $[\mathbb Q^+:S]<\infty$ means that there is $m>0$ and a finite partition of primes into sets $P_i$, $i<n$, such that $x\in S$ whenever $m\mid\sum_{p\in P_i}v_p(x)$ for all $i<n$. $\endgroup$ Commented Nov 18 at 16:00
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    $\begingroup$ Getting the index down to one requires ruling out that there is a nonconstant multiplicative function $\lambda$ with $\lambda(p-1)=1$ for all primes $p$. This is of course true but seems hard to prove. $\endgroup$ Commented Nov 18 at 18:59
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    $\begingroup$ @WillSawin Such a simple-to-state hard problem! I remark that for completely multiplicative functions, it would follow from the standard (hard) conjecture that the least prime congruent to 1 (mod $q$) is always less than $q^2$ (let $q$ be minimal with $\lambda(q)\ne1$...). $\endgroup$ Commented Nov 18 at 23:22
  • $\begingroup$ @GregMartin I actually meant a completely multiplicative function so your implication is unqualified. $\endgroup$ Commented Nov 19 at 1:53
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    $\begingroup$ @GregMartin It also follows from Schinzel's hypothesis (or just Dickson's conjecture) as noted in Zhi-Wei Sun's answer. $\endgroup$ Commented Nov 19 at 6:11