# Two remarks about Picard-Vessiot extensions and elementary functions

Colloquium Mathematicae (2000)

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

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topŻołądek, 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

top- [Bor] A. Borel, Linear Algebraic Groups, Benjamin, New York, 1969.
- [Dav] J. H. Davenport, On the Integration of Algebraic Functions, Springer, Berlin, 1981.
- [Kap] I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. Zbl0083.03301
- [Kol] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. Zbl0264.12102
- [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.
- [Mag] A. G. Magid, Lectures on Differential Galois Theory, Amer. Math. Soc., Providence, 1994.
- [Rit] J. F. Ritt, Integration in Finite Terms. Liouville's Theory of Elementary Methods, Columbia Univ. Press, New York, 1948.
- [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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