Galois Theory on the Line in Nonzero Characteristic

and let α1, . . . , αn be its roots, which are assumed to be distinct. By definition, the Galois Group G of this equation consists of those permutations of the roots which preserve all relations between them. Equivalently, G is the set of all those permutations σ of the symbols {1, 2, . . . , n} such that φ(ασ(1), . . . , ασ(n)) = 0 for every n-variable polynomial φ for which φ(α1, . . . , αn) = 0. The coefficients of φ are supposed to be in a field K which contains the coefficients a1, . . . , an of the given polynomial f = f(Y ) = Y n + a1Y n−1 + · · ·+ an. We call G the Galois Group of f over K and denote it by GalY (f,K) or Gal(f,K). This is Galois’ original concrete definition. According to the modern abstract definition, the Galois Group of a normal extension L of a field K is defined to be the group of all K-automorphisms of L and is denoted by Gal(L,K). Note that a normal extension L of a field K is a field obtained by adjoining to K all the roots of a bunch of univariate polynomials with