Abhyankar, Shreeram S. Pillars and towers of quadratic transformations. (English) Zbl 1227.14004 Proc. Am. Math. Soc. 139, No. 9, 3067-3082 (2011). A field \(k\) has the kroneckerian dimension \(n\) if its transcendence degree over its prime subfield is either \(n\) if char \(k>0\), or \(n+1\) otherwise. For any field \(k\) of kroneckerian dimension \(n\) there exists a subring \(A\) of \({\mathbb Z}[T_1,\ldots, T_{n+1}]\) such that \(k\) is a factor ring of \(A\). The proof uses the so called infinite pillars of quadratic transformations. Towers whose underlying quadratic transformations are finite pillars or nonpillars are employed for the construction of basic dicritical divisors. Reviewer: Dorin-Mihail Popescu (Bucureşti) Cited in 4 ReviewsCited in 10 Documents MSC: 14A05 Relevant commutative algebra 13H05 Regular local rings 13F30 Valuation rings 14C20 Divisors, linear systems, invertible sheaves Keywords:kroneckerian dimension of fields; quadratic transformations; towers; pillars; dicritical divisors × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321 – 348. · Zbl 0074.26301 · doi:10.2307/2372519 [2] Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. · Zbl 0093.04501 [3] S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. · Zbl 0914.14006 [4] S. S. Abhyankar, Lectures on algebra. Vol. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. · Zbl 1121.13001 [5] S. S. Abhyankar, Inversion and Invariance of Characteristic Terms. Part I, Developments in Mathematics, Springer, 2010. · Zbl 1322.14054 [6] Shreeram S. Abhyankar, Dicritical divisors and Jacobian problem, Indian J. Pure Appl. Math. 41 (2010), no. 1, 77 – 97. · Zbl 1197.14061 · doi:10.1007/s13226-010-0017-x [7] S. S. Abhyankar and W. J. Heinzer, Existence of dicritical divisors, to appear in American Journal of Mathematics. · Zbl 1248.13021 [8] S. S. Abhyankar and I. Luengo, Algebraic theory of dicritical divisors, to appear in American Journal of Mathematics. · Zbl 1232.13012 [9] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. · Zbl 0322.13001 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.