Rumely's local global principle for algebraic ${\mathrm P}{\mathcal S}{\mathrm C}$ fields over rings.

Let $ \mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $ \mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $ \sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $ \mathbb{Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $ \mathbb{Q}$ by $ \mathbb{Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $ \mathcal{V}_{M}$ (resp., $ \mathcal{S}_{M}$). If $v\in \mathcal{V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all $ {\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{S}_{M}$. Then $V(O_{M})\not =\varnothing $. We also prove two approximation theorems for $M$. [ABSTRACT FROM AUTHOR]