### Some remarks on the work of Jules Drach

It seems that a lot can be learnt, especially from the point of view of constructive mathematics, from the work of Jules Drach (1871-1941)

Mathematicians such as Hadamard, Lebesgue, Lusin and Borel were very impressed by his work. Borel even traced back his discovery of measure (see measure) to Drach's influence.

In a general introduction to algebra, Drach gives a presentation of Galois theory which seems quite relevant to constructive mathematics.

The viewpoint developed there gave rise both to measure theory and differential algebra. I will try now to describe it in more details.

The general problem is phrased as follows: we want to introduce -symbols- x1,...,xn submitted to some conditions

p1(x1,...,xn) = ... = pm(x1,...,xn) = 0 (*) where p1,...,pm are rational polynomials.

The first question is if this is -possible-. This is identified to the problem of whether the relations (*) are contradictory. That is: is there an extension L of Q where we can find x1,...,xn satisfying (*)?? More generally when is a relation p(x1,...,xn) = 0 a consequence of (*). Thus the problem of existence of a model is formulated as the problem of non contradiction of a theory.

The second question is if (*) characterises in a unique way the symbols x1,...,xn. If we have another relation r(x1,...,xn) = 0 which is compatible with (*), is it the case that this is a consequence of (*)??

It is remarkable that the first problem has the following solution (this result is a formal version of Hilbert Nullstellensatz): p = 0 is a consequence of (*) iff p is in the radical of the ideal generated by p1,...,pm. This results has a direct (elementary) proof.

The second problem becomes: is this radical prime?? If it is not we have some branching and we can add, in a non canonical way, a finite number of new relations so that they become prime (this corresponds to the fact that a radical ideal is a finite intersection of prime ideals).

This is a much nicer description of the Galois group of an equation than the usual one. For instance the Galois group of x^n - a1 x^(n-1) + ... = 0 is general iff the radical of the ideal generated by the relations

x1 + ... + xn = a1

x1 x2 + ... = a2

...

x1...xn = an

is prime.

At about the same time, in his thesis, Drach introduced the concept of differential algebra, and tried to develop Galois theory in this framework. Drach saw this work as a general theory of classification of ``transcendental quantities'' and thought that Galois had similar ideas (formulated in his last letter). A simple example: how do we justify the exponential function? We look at the differential equation

y' = y (**)

and we show that this equation is not contradictory. More generally, one can show that we cannot have as consequence of (**) an algebraic relation f(x,y) = 0. This establishes both the ``existence'' of the exponential function and the fact that this function is not an algebraic function.

It is remarkable also that Drach could foresee that the same method would apply to difference equations (cf. the work of Karr, JACM 81)