Differential Galois theory

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In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is e−x2, whose antiderivative is (up to constants) the error function, familiar from statistics. Other examples include the functions

\frac{\sin(x)}{x}

and

\quad x^x.

It should be realized 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 e−x2 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 that 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 that 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.

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[edit] 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.

[edit] 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.

[edit] 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 c1, ..., cn in Con(F), u1, ..., un, v in F such that

a = c_1\frac{Du_1}{u_1}+\dotsb+c_n\frac{Du_n}{u_n}+Dv.

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.

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