Talk:Differential Galois theory

I started this page in my user page, and was planning on doing quite a bit more work on it before moving it to a wikipedia article, but haven't really worked on it lately, so i thought that, in the spirit of the wiki, i will just move it now, in its unfinished state. I still plan on doing work on it, bringing it to a finished state, but in the meantime, here it is. Also, I now think that differential field extension and differential Galois theory should be separate articles (compare Galois theory vs. algebraic extension (or equivalently with field extension, which i think is a redundant article, and should be merged with algebraic extension)). Sooner or later someone should split this article. perhaps after it gets longer. right now, there is no Galois theory in this article, just some differential field theory. - Lethe 19:20, Jul 16, 2004 (UTC)

There are also a number of different 'flavours' of DGT. The 'textbook' one - automorphisms of differential fields - also seems to be one with which one can't do so much. But I do welcome having this page to launch the topic on WP.

Charles Matthews 19:24, 16 Jul 2004 (UTC)

I've added the See also section. I started writing something into the Risch algorithm page, which then required the elementary function page, after which I discovered this article. My knowledge of the field is limited: the Monthly paper, Moser's papers, Hardy, some ACM SIG stuff. I read this before any of the books in Computer Algebra came out. So change at will.   XaosBits 21:56, 10 May 2005 (UTC)

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Is it possible that there is a bug in the last section (Example...) ? it seems somehow rather uncomplete... — MFH: Talk 22:06, 10 May 2005 (UTC)

I've commented it out. If someone wants to add some content, and restore it, good job. As it stands now, it doesn't even get started:

== Example of theorem ==

Suppose we want to know whether a function of the form f*e^g has an elementary antiderivative, with f and g in C(x)

-GTBacchus 20:03, 9 November 2005 (UTC)

Differential algebra

I was somehow surprised to be redirected here from differential algebra. For me, differential algebra means an algebra with a derivation, e.g. lie algebras and so on. Thus this initial link should point to a page where derivations, not antiderivations, come first (or to the category of this name). — MFH: Talk 22:02, 10 May 2005 (UTC)

I redirected differential algebra to this article because I started writing an article on the subject and then discovered the Differential Galois theory article with the content I needed. The subject has its origins in Liouville's papers centered on a theorem (now known as Liouville's principle) that gives the form of an elementary integral. Differential algebra is being used in the sense of Joseph Ritt, whose book is available online from the AMS. Ritt's approach still has many analytical elements in it, but in 1968 Rosenlicht published an algebraic proof of Liouville's principle. Ritt seems to be well cited. The field of differential algebra has many applications for symbolic manipulations in programs such as Mathematica and Maple. As I noted, I'm a dilettante, so beware.   XaosBits 02:35, 11 May 2005 (UTC)

How is this defended?

'However, no matter how long the list of so called elementary functions, there will still be functions on the list whose antiderivatives are not."--VKokielov 05:02, 28 July 2005 (UTC)

If I put all continuous functions on the "list", then every function on the list surely has an antiderivative on the list. So, the statement is either nonsense, or it needs clarification of what is meant by list, and what functions are allowed to be included. — Emil J. 14:41, 8 October 2008 (UTC)

The List would have to be of infinite length, and would therefore never be finished.

Nevertheless, the statement as is stands is not exactly true and is of no consequence -- it talks about finite sets of functions, so they cannot even contain all complex constant functions -- also, it's false if the set contains only the 0 function (constant, equal to 0). I assume the idea was to say that if we include all constants and a finite number of functions together with all possible expressions made from them, sums, products then, given the set contains at least some functions we usually consider elemenentary, the consequence holds. I have no idea what this required subset is and don't know whether it's at all correct, so I'll refrain from changing the article. 83.9.165.125 (talk) 18:38, 9 February 2009 (UTC)

Example?

Hello! I am looking for an example of how to use the basic theorem. Is there an easy application to, for instance, prove that $e^{e^{x}}$ or $e^{x^{2}}$ doesn't have an elementary integral? Thanks A5 16:33, 14 February 2007 (UTC)

However, no matter how long the list of so called elementary functions, as long as it is finite, there will still be functions on the list whose antiderivatives are not.

This is not true. If we consider only polynomials to be elementary, any integral of elementary function will be elementary.--MathFacts (talk) 20:43, 11 November 2009 (UTC)