Skolem density problems over large Galois extensions of global fields. Appendix by Wulf-Dieter Geyer (Q2715536)

Let \(K\) be a global field. Fix a separable closure \(K_s\) and an algebraic closure \(\widetilde K\) of \(K\). Denote the set of all primes of \(K\) by \({\mathbb P}\), all finite primes by \({\mathbb P}_0\), all infinite primes by \({\mathbb P}_{\infty}\). For each \(\mathfrak p\in {\mathbb P}\) choose a separable algebraic extension \(K_{\mathfrak p}\) as follows. If \(\mathfrak p\in {\mathbb P}_0\), then \(K_{\mathfrak p}\) is a Henselization of \(K\) at \(\mathfrak p\). If \(\mathfrak p\in {\mathbb P}_{\infty}\) is real, then \(K_{\mathfrak p}\) is a real closure of \(K\) at \(\mathfrak p\). If \(\mathfrak p\in{\mathbb P}_{\infty}\) is complex, then \(K_{\mathfrak p}=K_s\). For \({\pmb\sigma}=({\sigma}_1,\ldots,{\sigma}_e)\in\text{Gal}(K)^e\) let \(K_s({\pmb\sigma})=\{x\in K_s\mid {\sigma}_i(x)=x,\;i=1,\dots,e\}\). Denote the maximal Galois extension of \(K\) inside \(K_s({\pmb\sigma})\) by \(K_s[{\pmb\sigma}]\). Fix an infinite proper subset \({\mathcal V}\) of \({\mathbb P}\), and a finite subset \(\mathcal S\) of \(\mathcal V\). Let NEWLINE\[NEWLINEK_{\text{tot},\mathcal S}=\bigcap_{{\mathfrak p}\in {\mathcal S}}\bigcap_{\sigma\in \text{Gal}(K)}K_{\mathfrak p}^{\sigma}NEWLINE\]NEWLINE be the field of totally \(\mathcal S\)-adic numbers.NEWLINENEWLINENEWLINEFor each \(\mathfrak p\in {\mathbb P}\) choose an absolute value \(||_{\mathfrak p}\) of \(K\) representing \(\mathfrak p\) and, for each prime \({\mathfrak q}\) of \(\widetilde K\) lying over \(\mathfrak p\), denote the absolute value of \(\widetilde K\) which represents \({\mathfrak q}\) and extends \(||_{\mathfrak p}\) by \(||_{\mathfrak q}\).NEWLINENEWLINENEWLINEAn algebraic extension \(M\) of \(K\) is called an \(\mathcal S\)-Skolem field with respect to \({\mathcal V}\) if the following holds: Let \(\mathcal T\) be a finite subset of \({\mathcal V}\) containing \(\mathcal S\). Put \(\mathcal U={{\mathbb P}_0}\cap({\mathcal V}\setminus\mathcal T)\). Let \(f_1,\ldots,f_m\) be \(\mathfrak q\)-primitive polynomials for each \(\mathfrak q\in\widetilde{\mathcal U}\). A polynomial is called \(\mathfrak q\)-primitive if its coefficients are \(\mathfrak q\)-integrals and at least one of them is a \(\mathfrak q\)-unit. Let \({\mathbf a}=(a_1,\ldots,a_n)\in M^n\) and \(\gamma >0.\) Then there exists \(x\in M^n\), with \(|x-a|_{\mathfrak p}<\gamma\) for each \(\mathfrak p\in\widetilde {\mathcal T}\), and \(|x|_{\mathfrak q}\leq 1, |f_i(\mathbf x)|_{\mathfrak q}=1\) for each \({\mathfrak q}\in\widetilde{\mathcal U}\), \(i=1,\dots,m\).NEWLINENEWLINENEWLINEThe goal of the paper under review is to prove the following Theorem A: Let \(e\) be a nonnegative integer. Then both \(K_s({\pmb\sigma})\cap K_{\text{tot},\mathcal S}\) and \(K_s[{\pmb\sigma}]\cap K_{\text{tot},\mathcal S}\) are \(\mathcal S\)-Skolem fields with respect to \(\mathcal V\), for almost all \({\pmb\sigma}\in \text{Gal}(K)^e\).NEWLINENEWLINENEWLINEThe theorem improves the main result of \textit{M. Jarden} and \textit{A. Razon} [Nieuw Arch. Wiskd. (4) 13, 381-399 (1995; Zbl 0859.12005)]. The theorem also generalizes a result of Skolem: Let \(g\in {\mathbb Z}[x]\) be a primitive polynomial. Then there is an algebraic integer \(x\) such that \(g(x)\) is an algebraic unit [\textit{Th. Skolem}, Skr. Norske Vid. Akad., Oslo 1935, No. 10, 1-19 (1935; Zbl 0011.19701)].NEWLINENEWLINEFor the entire collection see [Zbl 0955.00034].