property. The formal derivative satisfies the Leibniz rule: Note the order of the factors; when R is not commutative this is important. These two properties make D a derivation on A (see module of relative differential forms for a discussion of a generalization). Application to finding repeated factors As in calculus, the derivative detects multiple roots. If R is a field then R[x] is a Euclidean domain, and in this situation we can define multiplicity of roots; for every polynomial f(x) in R[x] and every element r of R, there exists a nonnegative integer mr and a polynomial g(x) such that where g(r)≠0. mr is the multiplicity of r as a root of f. It follows from the Leibniz rule that in this situation, mr is also the number of differentiations that must be performed on f(x) before r is no longer a root of the resulting polynomial. The utility of this observation is that although in general not every polynomial of degree n in R[x] has n roots counting multiplicity (this is the maximum, by the above theorem), we may pass to field extensions in which this is true (namely, algebraic closures). Once we do, we may uncover a multiple root that was not a root at all simply over R. For example, if R is the field with three elements, the polynomial has no roots in R; however, its formal derivative () is zero (why ?) since 3 = 0 in R and in any extension of R, so when we pass to the algebraic closure it has a multiple root that could not have been detected by factorization in R itself. Thus, formal differentiation allows an effective notion of multiplicity. This is important in Galois theory, where the distinction is made between separable field extensions (defined by polynomials with no multiple roots) and inseparable ones. Correspondence to analytic derivative When the ring R of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus. The element Y–X of the ring R[X,Y] divides Yn – Xn for any nonnegative integer n, and therefore divides f(Y) – f(X) for any polynomial f in one indeterminate. If the quotient in R[X,Y] is denoted by g, then It is then not hard to verify that g(X,X) (in R[X]) coincides with the formal derivative of f as it was defined