Let KqKq be the algebraic closure of the field FqFq (and so of FqmFqm). Let F(X)=g(X)−tF(X)=g(X)−t, where t PD 0332991 an indeterminate. For given a¯, Ga¯ denotes the Galois group of g(X)−tg(X)−t over Fqm(t)Fqm(t), where t is an indeterminate. It has as a normal subgroup Gˆa¯, the Galois group of g(X)−tg(X)−t over Kq(t)Kq(t). An important criterion for Gˆa¯ to be the full symmetric group SnSn derives from Theorem 4.8 of [7].
Lemma 5.1.
Let g(X)∈Fq[X]g(X)∈Fq[X]be monic of degree n and indecomposable over FqFq(i.e., g is cytosine not a composition g=g1(g2)g=g1(g2)of polynomials g1(X),g2(X)∈Fq[X]g1(X),g2(X)∈Fq[X], where deg?(gi)≥2deg?(gi)≥2, i=1,2i=1,2). Suppose that, for some θ∈Kqθ∈Kq, g(X)−θg(X)−θfactorizes over KqKqas (X−β)2E(X)(X−β)2E(X)for some square-free polynomial E (with E(β)≠0E(β)≠0). Then the Galois group of g(X)−tg(X)−tover Kq(t)Kq(t)is SnSn.
Lemma 5.1.
Let g(X)∈Fq[X]g(X)∈Fq[X]be monic of degree n and indecomposable over FqFq(i.e., g is cytosine not a composition g=g1(g2)g=g1(g2)of polynomials g1(X),g2(X)∈Fq[X]g1(X),g2(X)∈Fq[X], where deg?(gi)≥2deg?(gi)≥2, i=1,2i=1,2). Suppose that, for some θ∈Kqθ∈Kq, g(X)−θg(X)−θfactorizes over KqKqas (X−β)2E(X)(X−β)2E(X)for some square-free polynomial E (with E(β)≠0E(β)≠0). Then the Galois group of g(X)−tg(X)−tover Kq(t)Kq(t)is SnSn.