  In this paper the transcendental Galois extensions of a field will be
introduced as counterparts to algebraic Galois ones. There exist several types
of transcendental Galois extensions of a given field, from the weakest one to
the strongest one, such as Galois, tame Galois, strong Galois, and absolute
Galois. The four Galois extensions are distinct from each other in general but
coincide with each other for cases of algebraic extensions. The transcendental
Galois extensions arise from higher relative dimensional Galois covers of
arithmetic schemes. In the paper we will obtain several properties of Galois
extensions in virtue of conjugation and quasi-galois and will draw a comparison
between the Galois extensions. Strong Galois is more accessible. It will be
proved that a purely transcendental extension is strong Galois.
