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Galois theory on the line in nonzero characteristic. (English) Zbl 0760.12002

This is a fine paper, which should be read and enjoyed. Let \(f=y^ n+a_ 1y^{n-1}+\cdots+a_ n\) be a polynomial over a field \(K\) and let \(\text{Gal}(f,K)\) be the Galois group of \(f\) over \(K\). In 1957, the author defined the algebraic fundamental group \(\Pi_{A(C)}\) for a nonsingular irreducible (open) curve \(C\) defined over an algebraically closed field \(k\) and stated some conjectures regarding \(\Pi_ A(L_{k,r})\). In that connection, in support of these conjectures, he stated the following two problems:
Let \(0=a\in K\), \(K\) an algebraically closed field of characteristic \(p\neq 0\) and let \(n,s,t\) be positive integers such that \(t\not\equiv 0(p)\). Let \(\hat F_ n=y^ n-ax^ sy^ t+1\) with \(n=p+t\) and \(\tilde F_ n=y^ n-ay^ t+x^ s\) with \(t<n\equiv 0(p)\) and GCD\((n,t)=1\) and \(s\equiv 0(t)\). Let \(\hat G_ n=\text{Gal}(\hat F_ n,K(x))\) and \(\tilde G_ n=\text{Gal}(\tilde F_ n,K(x))\). The problem is how to calculate \(\hat G_ n\) and \(\tilde G_ n\).
The author calculates these groups (after a gap of 30 years). The case \(t=1\) was proved by Serre. The other 8 cases are due to the author.
Many important corollaries follow settling some conjectures.

MSC:

12F10 Separable extensions, Galois theory
14H30 Coverings of curves, fundamental group
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