Advertisement

Mathematical Notes

, Volume 86, Issue 5–6, pp 625–628 | Cite as

On the saturation of subfields of invariants of finite groups

  • I. V. Arzhantsev
  • A. P. Petravchuk
Article
  • 50 Downloads

Abstract

Every subfield \( \mathbb{K} \)(φ) of the field of rational fractions \( \mathbb{K} \)(x 1,..., x n ) is contained in a unique maximal subfield of the form \( \mathbb{K} \)(ω). The element ω is said to be generating for the element φ. A subfield of \( \mathbb{K} \)(x 1,..., x n ) is said to be saturated if, together with every its element, the subfield also contains the generating element. In the paper, the saturation property is studied for the subfields of invariants \( \mathbb{K} \)(x 1,..., x n ) G of a finite group G of automorphisms of the field \( \mathbb{K} \)(x 1..., x n ).

Key words

finite group saturated subfield polynomial ring polynomial invariant subalgebra of invariants closed rational function the groups SL2(ℂ) PSL2(ℂ). 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. V. Arzhantsev and A. P. Petravchuk, “Closed polynomials and saturated subalgebras of polynomial algebras,” UkrainianMath. J. 59(12), 1783–1790 (2007).CrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Moulin Ollagnier, “Algebraic closure of a rational function,” Qual. Theory Dyn. Syst. 5(2), 285–300 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. P. Petravchuk and O. G. Iena, “On closed rational functions in several variables,” Algebra Discrete Math., No. 2, 115–124 (2007).Google Scholar
  4. 4.
    A. van den Essen, J. Moulin Ollagnier, and A. Nowicki, “Rings of constants of the form k[f],” Comm. Algebra 34(9), 3315–3321 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Bodin, “Reducibility of rational functions in several variables,” Israel J. Math. 164, 333–347 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and Applications of Finite Groups (Dover Publ., New York, NY, 1916).zbMATHGoogle Scholar
  7. 7.
    S. S.-T. Yau and Y. Yu, Gorenstein Quotient Singularities in Dimension Three, inMem. Amer. Math. Soc. (Amer. Math. Soc., Providence, RI, 1993), Vol. 105, no. 505.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Kiev National UniversityKievUkraine

Personalised recommendations