Let $f:X\to Y$ be a surjective morphism of integral schemes. Then $X$ is said
to be quasi-galois closed over $Y$ by $f$ if $X$ has a unique conjugate over
$Y$ in an algebraically closed field. Such a notion has been applied to the
computation of \'{e}tale fundamental groups.

In this paper we will use affine coverings with values in a fixed field to
discuss quasi-galois closed and then give a sufficient and essential condition
for quasi-galois closed. Here, we will avoid using affine structures on a
scheme since their definition looks copious and fussy.

Passing from arithmetic schemes to algebraic schemes, in a similar manner we
will have the computation of the \'{e}tale fundamental group of an algebraic
scheme and then will define and discuss the qc fundamental group of an
algebraic scheme in this paper. The qc fundamental group will also give a prior
estimate of the \'{e}tale fundamental group.

In this paper we will give the computation of the \'etale fundamental group
of an arithmetic scheme.

In this paper we will give an explicit construction of the geometric model
for a prescribed extension of a function field in several variables over a
number field.

As a by-product, we will also prove the existence of quasi-galois closed
covers of arithmetic schemes (in eprint arXiv:0907.0842).