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Alternating group coverings of the affine line for characteristic greater than two. (English) Zbl 0801.12003

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.

MSC:

12F10 Separable extensions, Galois theory
14H30 Coverings of curves, fundamental group
20D06 Simple groups: alternating groups and groups of Lie type
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References:

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