## 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

### Keywords:

Galois group; algebraic fundamental group
Full Text:

### References:

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