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Linear disjointness of polynomials. (English) Zbl 0760.12003

Let \(k\) be an algebraically closed field, and let there be given any two nonconstant monic separable polynomials \(f(X,Y)\) and \(g(X,Y)\) in \(Y\) with coefficients in \(k(X)\). The author proves the following result: For most \(a,b\) in \(k\) with \(a\neq 0\), the splitting fields of the polynomials \(f(X,Y)\) and \(g(aX+b,Y)\) over \(k(X)\) are linearly disjoint over \(k(X)\).
The above result rests on the following lemma proved by the author: Let \(K\) be a finitely generated field extension of \(k\) of transcendence degree one, and let \(x\in K\) with \(x\notin k\). Assume that either the genus of \(K/k\) is nonzero or \(K/k(x)\) is not purely inseparable. Then for most \(a,b\) in \(k\), we have \(ax+b\notin\{\tau(x)|\tau\in\text{Aut}_ kK\}\) where \(\text{Aut}_ kK\) is the group of all automorphisms of \(K\). — The author also deduces that the algebraic fundamental group of an affine line is closed relative to direct products.

MSC:

12F10 Separable extensions, Galois theory
14H30 Coverings of curves, fundamental group
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[1] Shreeram S. Abhyankar, Wreath products and enlargements of groups, Discrete Math. 120 (1993), no. 1-3, 1 – 12. · Zbl 0787.20019 · doi:10.1016/0012-365X(93)90560-G
[2] Kenkichi Iwasawa and Tsuneo Tamagawa, On the group of automorphisms of a function field, J. Math. Soc. Japan 3 (1951), 137 – 147. · Zbl 0044.26901 · doi:10.2969/jmsj/00310137
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