From the introduction: Resolution of singularities of surfaces over the field of complex numbers was first achieved by the J. E. Jung in 1908. Jung’s proof was based on the fact that the local fundamental group above a normal crossing of the branch locus is abelian. In 1955 [S. Abbyyankar, Am. J. Math. 77, 575-592 (1955; Zbl 0064.27501)] it was shown that, in characteristic \(p>0\), such a local fundamental group may not even be solvable. In 1975 [S. Abhyankar, ibid. 79, 825-856 (1957; Zbl 0087.03603)], this led to the conjecture that the algebraic fundamental group of the affine line over an algebraically closed ground field of characteristic \(p>0\) coincides with \(Q(p)\), where \(Q(p)\) is the set of all quasi-\(p\) groups, i.e., finite groups generated by their \(p\)-Sylow subgroups. In the 1957 paper it was also conjectured that the algebraic fundamental group of the (once) punctured affine line over an algebraically closed ground field of characteristic \(p>0\) coincides with \(Q_1(p)\), where \(Q_1(p)\) is the set of all cyclic-by-quasi-\(p\) groups, i.e., finite groups \(G\) such that \(G/p(G)\) is cyclic where \(p(G)\) is the subgroup of \(G\) generated by all of its \(p\)-Sylow subgroups. In support of these conjectures, in the 1957 paper, two families of equations were written down giving unramified coverings of the affine line and the punctured affine line, and it was suggested that their Galois groups be computed. These families were originally obtained by taking a section of the unsolvable surface covering of the 1955 paper.
In section 3 and 6, it now turns out that the Galois groups of these families include the alternating and symmetric groups \(A_n\) and \(S_n\) of any degree \(n>1\), the linear groups \(\text{GL}(m,q)\), \(\text{SL}(m,q)\), \(\text{PGL}(m,q)\), \(\text{PSL}(m,q)\), \(\text{AGL}(1,q)\) and \(\text{GF}(q)^+\) for any prime power \(q>1\) and any integer \(m>1\), the Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), and \(M_{24}\), and the group \(\operatorname{Aut}(M_{12})\).
In particular, the trinomial equation \(Y^{23}-XY^t+1=0\) has Galois group \(M_{23}\) for \(p=2\) and \(t=3\). In section 4, it is shown that for all other relevant values of \(p\) and \(t\) its Galois group is \(A_{23}\) or \(S_{23}\).
In section 5, a merging technique is explained which shows that, in the trinomial case, sometimes the two families can be converted into each other. In section 6, this together with reciprocation, leads to a list of equations having Mathieu groups as Galois groups. In section 7, there are proved some transitivity lemmas. In section 8, these are used to show that some other members of the first family also give unramified coverings of the affine line and the punctured affine line with Galois groups \(\text{GL}(m,q)\), \(\text{SL}(m,q)\), \(\text{PGL}(m,q)\) and \(\text{PSL}(m,q)\), and as a consequence there are deduced several explicit equations which have these groups as Galois groups and which give unramified coverings of the \(m\)-dimensional affine space or the \(m\)-dimensional affine space minus a hyperplane.
For the entire collection see [Zbl 0823.00012].