Let K be a global field, V a proper subset of the set of all primes of K, S a finite subset of V, and \tilde K (resp. K_sep) a fixed algebraic (resp. separable algebraic) closure of K. Let Gal(K)=Gal(K_sep/K) be the absolute Galois group of K. For each p∈V we choose a Henselian (respectively, a real or algebraic) closure K_p of K at p in \tilde K if p is nonarchimedean (respectively, archimedean). Then, K_tot,S=\cap_p∈S\cap_τ∈Gal(K)K_p^τ is the maximal Galois extension of K in K_sep in which each p∈S totally splits. For each p∈V we choose a p-adic absolute value | |_p of K_p and extend it in the unique possible way to \tilde K.

For σ=(σ₁,...,σ_e)∈Gal(K)^e let K_tot,S[σ] be the maximal Galois extension of K in K_tot,S fixed by σ₁,...,σ_e. Then, for almost all σ∈Gal(K)^e (with respect to the Haar measure), the field K_tot,S[σ] satisfies the following local-global principle:

Let V be an absolutely integral affine variety in A_Kⁿ. Suppose that for each p∈S there exists z_p∈ V_simp(K_p) and for each p∈V\S there exists z_p∈ V(\tilde K) such that in both cases |z_p|_p≦1 if p is nonarchimedean and |z_p|_p<1 if p is archimedean. Then, there exists z∈ V(K_tot,S[σ]) such that for all p∈V and for all τ∈Gal(K) we have: |z^τ|_p≦1 if p is archimedean and |z^τ|_p<1 if p is nonarchimedean.