Abhyankar, Shreeram S. Linear disjointness of polynomials. (English) Zbl 0760.12003 Proc. Am. Math. Soc. 116, No. 1, 7-12 (1992). 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. Reviewer: J.N.Mordeson (Omaha) Cited in 1 ReviewCited in 4 Documents MSC: 12F10 Separable extensions, Galois theory 14H30 Coverings of curves, fundamental group Keywords:separable polynomials; splitting fields; algebraic fundamental group PDFBibTeX XMLCite \textit{S. S. Abhyankar}, Proc. Am. Math. Soc. 116, No. 1, 7--12 (1992; Zbl 0760.12003) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.