Ukrainian Mathematical Journal

, 59:1783 | Cite as

Closed polynomials and saturated subalgebras of polynomial algebras

  • I. V. Arzhantsev
  • A. P. Petravchuk


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


Generative Element Generative Polynomial Irreducible Polynomial Polynomial Algebra Integral Closure 
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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • I. V. Arzhantsev
    • 1
  • A. P. Petravchuk
    • 2
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Shevchenko Kiev National UniversityKievUkraine

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