Two remarks about Picard-Vessiot extensions and elementary functions

• Autorzy: Henryk Żołądek
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 84/85
• Numer: 1
• Strony: 173-183

Daty

wydano

2000

otrzymano

1999-06-11

poprawiono

1999-10-01

Twórcy

autor

Henryk Żołądek

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [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.
• [Kol] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
• [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.