Let K be a global field,
$\mathcal{V}$
a proper subset of the set of all primes of K,
$\mathcal{S}$
a finite subset of
$\mathcal{V}$
, and
${\tilde K}$
(resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with
$K_\mathrm{sep}\{\subseteq}{\tilde K}$
. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each
$\mathfrak{p}\in\mathcal{V}$
, we choose a Henselian (respectively, a real or algebraic) closure
$K_\mathfrak{p}$
of K at
$\mathfrak{p}$
in
${\tilde K}$
if
$\mathfrak{p}$
is non-archimedean (respectively, archimedean). Then,
$K_{\mathrm{tot},\mathcal{S}}=\bigcap_{\mathfrak{p}\in\mathcal{S}}\bigcap_{\tau\in{\rm Gal}(K)}K_\mathfrak{p}^\tau$
is the maximal Galois extension of K in Ksep in which each
$\mathfrak{p}\in\mathcal{S}$
totally splits. For each
$\mathfrak{p}\in\mathcal{V}$
, we choose a
$\mathfrak{p}$
-adic absolute value
$|~|_\mathfrak{p}$
of
$K_\mathfrak{p}$
and extend it in the unique possible way to
${\tilde K}$
. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in
$\mathbb{A}_K^n$
. Suppose that for each
$\mathfrak{p}\in\mathcal{S}$
there exists a simple
$K_\mathfrak{p}$
-rational point
$\mathbf{z}_\mathfrak{p}$
of V and for each
$\mathfrak{p}\in\mathcal{V}\smallsetminus\mathcal{S}$
there exists
$\mathbf{z}_\mathfrak{p}\in V({\tilde K})$
such that in both cases
$|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}\le1$
if
$\mathfrak{p}$
is non-archimedean and
$|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}<1$
if
$\mathfrak{p}$
is archimedean. Then, there exists
$\mathbf{z}\in V(K_{\mathrm{tot},\mathcal{S}}\cap K_\mathrm{symm})$
such that for all
$\mathfrak{p}\in\mathcal{V}$
and for all τ ∈ Gal(K), we have
$|\mathbf{z}^\tau|_\mathfrak{p}\le1$
if
$\mathfrak{p}$
is archimedean and
$|\mathbf{z}^\tau|_\mathfrak{p}<1$
if
$\mathfrak{p}$
is non-archimedean. For
$\mathcal{S}=\emptyset$
, we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.