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