DIFFERENTIAL GALOIS THEORY

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Contents

Motivation and basic idea

Definitions

Examples of defined terms

Basic theorem

External links

See also

Motivation and basic idea

In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. The most often encountered example of such a function is $e^\{-x^2\}$, whose antiderivative is (up to constants) the error function, familiar from statistics. Other examples include and $x^x$.
It should be realised that the notion of an elementary function is merely a matter of convention. One could choose to add the error function to the list of elementary functions, and with this new list, the antiderivative of exp(-''x''²) is elementary. However, no matter how long the list of so called elementary functions, there will still be functions on the list whose antiderivatives are not.
The machinery of 'differential Galois theory' allows one to determine when an elementary function does or does not have an antiderivative which can be expressed as an elementary function. Differential Galois theory is a theory based on the model of Galois theory. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields which are equipped with a derivation, ''D''. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.

Definitions

For any differential field ''F'', there is a subfield
:Con(''F'') = {''f'' in ''F'' | ''Df'' = 0},
called the constants of ''F''. Given two differential fields ''F'' and ''G'', ''G'' is called a 'logarithmic extension' of ''F'' if ''G'' is a simple transcendental extension of ''F'' (i.e. ''G'' = ''F''(''t'') for some transcendental ''t'') such that
:''Dt'' = ''Ds''/''s'' for some ''s'' in ''F''.
This has the form of a logarithmic derivative. Intuitively, one may think of ''t'' as the logarithm of some element ''s'' of ''F'', in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that ''F'' is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to ''F''. Similarly, an 'exponential extension' is a simple transcendental extension which satisfies
:''Dt'' = ''tDs''.
With the above caveat in mind, this element may be thought of as an exponential of an element ''s'' of ''F''. Finally, ''G'' is called an 'elementary differential extension' of ''F'' if there is a finite chain of subfields from ''F'' to ''G'' where each extension in the chain is either algebraic, logarithmic, or exponential.

Examples of defined terms

As an example, the field 'C'(''x'') of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers 'C'.

Basic theorem

Suppose ''F'' and ''G'' are differential fields, with Con(''F'') = Con(''G''), and that ''G'' is an elementary differential extension of ''F''. Let ''a'' be in ''F'', ''y'' in G, and suppose ''Dy'' = ''a'' (in words, suppose that ''G'' contains an antiderivative of ''a''). Then there exist ''c''₁, ..., ''c''_n in Con(''F''), ''u''₁, ..., ''u''_''n'', ''v'' in ''F'' such that
:
In other words, the only functions that have "elementary antiderivatives" (i.e. antiderivatives living in, at worst, an elementary differential extension of ''F'') are those with this form prescribed by the theorem. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.

External links

★ Differential Galois Theory, M. van der Put and M. F. Singer

See also

★ Risch algorithm

★ Elementary functions