A constructive version of Laplace’s proof on the existence of complex roots

Gauss presented several proofs that the field complex numbers is algebraically closed. His second proof [3] is sometimes described as a rigorous version of a previous proof due to Laplace [6]. Both proofs indeed can be seen as arguments showing that if R is a real closed field then the field C = R[i] obtained by adding a root of i2 + 1 = 0 is algebraically closed. Theses two proofs are however different: the proof of Gauss for instance refers to the notion of discriminant of a polynomial, which is not used in Laplace’s proof. Laplace’s argument is interesting from the point of view of constructive mathematics since it relies on the existence of a splitting field of an arbitrary nonconstant polynomial in C[X]. The existence of such a splitting field does not raise any problem from a classical point of view, and it is interesting that Gauss criticizes Laplace’s argument on this point1. One analysis of the notion of splitting field from a constructive point of view can be found in Edwards’ book [2]. But this analysis relies on a factorization algorithm, which exists only in special (but important) cases: for instance over rational numbers, or over fields of the form Q(X1, . . . , Xn) or over algebraic extensions of such fields. The classic book of van der Waerden [10] presents such algorithms, and the reference [9] refines this analysis. For a general discrete field, however, we cannot hope for a factorization algorithm [10, 8]. We apply here a different constructive analysis, inspired by some remarks of A. Joyal [5]. We use this to present a constructive version of Laplace’s proof, different from the proof of Gauss. As we said, this is an application of a general method for making constructive sense of the notion of a splitting field of a nonconstant polynomial over an arbitrary field, and we present this method in the second part of this paper.