Alternating group coverings of the affine line for characteristic greater than two
Let \(L_ k\) be the affine line over an algebraically closed ground field \(k\) of nonzero characteristic \(p\), let \(q\) be a positive power of \(p\) and let \(n\), \(t\) be two integers with \(n= q+t\), \(t\not\equiv 0\pmod p\). Let the unramified covering of \(L_ k\) be given by \(\overline{F}_{n,q} =0\), where \(\overline{F}_{n,q}= Y^ n- XY^ t +1\) and let \(\overline{G}_{n,q}= \text{Gal} (\overline{F}_{n,q}, k(X))\) be the associated Galois group. By using some arguments about the permutation groups \(A_ n\) (alternating group), \(S_ n\) (symmetric group) and the relations with the Galois theory, the author establishes the main theorems of the paper: Theorem 1. If \(p>2\) and \(q<t\), then \(\overline{G}_{n,q}= A_ n\); Theorem 2. If \(p=q =2\) then \(\overline{G}_{n,q}= S_ n\). If \(2= p<q <t\) then \(\overline{G}_{n,q}= A_ n\) or \(S_ n\). The paper, which also contains other interesting theorems, completes a series of works of the author concerning the subject.
scientific article; zbMATH DE number 537286
scientific article