Semilinear transformations

In previous papers, nice trinomial equations were given for unramified cove rings of the once punctured affine line in nonzero characteristic p with th e projective general group PGL(m, q) and the general linear group GL(m, q) as Galois groups where m >1 is any integer and q >1 is any power of p. Thes e Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups w e get the projective general semilinear group PGL(m, q) and the general sem ilinear group Gamma L(m,q). We also obtain the semilinear versions of the l ocal coverings considered in previous papers.