Closed polynomials and saturated subalgebras of polynomial algebras - Arzhantsev, I. V.; Petravchuk, A. P. (Ukrains’kyi Matematychnyi Zhurnal, 2007)
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.