Arzhantsev, I. V.; Petravchuk, A. P. Closed polynomials and saturated subalgebras of polynomial algebras. (English) Zbl 1164.13302 Ukr. Mat. Zh. 59, No. 12, 1587-1593 (2007) and Ukr. Math. J. 59, No. 12, 1783-1790 (2007). Summary: The behaviour of closed polynomials, i.e., polynomials \(f\in\mathbb{K}[x_1,\dots,x_n]\setminus\mathbb{K}\) such that the subalgebra \(\mathbb{K}[f]\) is integrally closed in \(\mathbb{K}[x_1,\dots,x_n]\), is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial \(f\in\mathbb{K}[x_1,\dots,x_n]\setminus\mathbb{K}\) can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras \(A\subset\mathbb{K}[x_1,\dots,x_n]\), i.e., subalgebras such that, for any \(f\in A\setminus\mathbb{K}\), a generative polynomial of \(f\) is contained in \(A\). Cited in 2 ReviewsCited in 11 Documents MSC: 13B25 Polynomials over commutative rings Keywords:closed polynomials; subalgebra; polynomial algebra PDFBibTeX XMLCite \textit{I. V. Arzhantsev} and \textit{A. P. Petravchuk}, Ukr. Mat. Zh. 59, No. 12, 1587--1593 (2007; Zbl 1164.13302) Full Text: DOI Link