Two remarks about Picard-Vessiot extensions and elementary functions

Henryk Żołądek

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 173-183
  • ISSN: 0010-1354

Abstract

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We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group G a l L M is a normal subgroup of G a l K M . We also present a proof that the probability function Erf(x) is not an elementary function.

How to cite

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Żołądek, Henryk. "Two remarks about Picard-Vessiot extensions and elementary functions." Colloquium Mathematicae 84/85.1 (2000): 173-183. <http://eudml.org/doc/210795>.

@article{Żołądek2000,
abstract = {We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group $Gal_\{L\} M$ is a normal subgroup of $Gal_\{K\} M$. We also present a proof that the probability function Erf(x) is not an elementary function.},
author = {Żołądek, Henryk},
journal = {Colloquium Mathematicae},
keywords = {differential Galois theory},
language = {eng},
number = {1},
pages = {173-183},
title = {Two remarks about Picard-Vessiot extensions and elementary functions},
url = {http://eudml.org/doc/210795},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Żołądek, Henryk
TI - Two remarks about Picard-Vessiot extensions and elementary functions
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 173
EP - 183
AB - We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group $Gal_{L} M$ is a normal subgroup of $Gal_{K} M$. We also present a proof that the probability function Erf(x) is not an elementary function.
LA - eng
KW - differential Galois theory
UR - http://eudml.org/doc/210795
ER -

References

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  1. [Bor] A. Borel, Linear Algebraic Groups, Benjamin, New York, 1969. 
  2. [Dav] J. H. Davenport, On the Integration of Algebraic Functions, Springer, Berlin, 1981. 
  3. [Kap] I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. Zbl0083.03301
  4. [Kol] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. Zbl0264.12102
  5. [Lio] J. Liouville, Premier mémoire sur la détermination des intégrales dont la valeur est algébrique, J. École Polytech. 14 (1833), 124-148; Second mémoire sur la détermination des intégrales dont la valeur est algébrique, ibid., 149-193. 
  6. [Mag] A. G. Magid, Lectures on Differential Galois Theory, Amer. Math. Soc., Providence, 1994. 
  7. [Rit] J. F. Ritt, Integration in Finite Terms. Liouville's Theory of Elementary Methods, Columbia Univ. Press, New York, 1948. 
  8. [Sin] M. F. Singer, Algebraic relations among solutions of linear differential equations, Trans. Amer. Math. Soc. 295 (1986), 753-763. Zbl0593.12014

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