# Closed polynomials and saturated subalgebras of polynomial algebras

### Abstract

The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$.
Published

25.12.2007

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 59, no. 12, Dec. 2007, pp. 1587–1593, https://umj.imath.kiev.ua/index.php/umj/article/view/3414.

Issue

Section

Research articles