Formal derivative

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In mathematics, the formal derivative is an operation on elements of a polynomial ring which mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general impossible to define for a ring. Many of the properties of the derivative are true of the formal derivative, but some, especially those which make numerical statements, are not. The primary use of formal differentiation in algebra is to test for multiple roots of a polynomial.

The definition of a formal derivative is as follows: fix a ring R (not necessarily commutative) and let A = R[x] be the ring of polynomials over R. Then the formal derivative is an operation on elements of A, where if

[f(x)\,=\,a_n x^n + \dots + a_1 x + a_0]

then its formal derivative is

[f'(x)\,=\,Df(x) = n a_n x^ + \dots + 2 a_2 x + a_1]

just as for polynomials over the real or complex numbers. It can be verified that:

Formal differentiation is linear: for any two polynomials f(x), g(x) and elements r, s of R, we have

[(r \cdot f + s \cdot g)'(x) = r \cdot f'(x) + s \cdot g'(x)]

When R is not commutative there is another, different linearity property in which r and s appear on the right rather than on the left. When R does not contain an identity element then neither of these reduces to the case of simply a sum of polynomials or the sum of a polynomial with a multiple of another polynomial, which must also be included as a "linearity" property.

The formal derivative satisfies the Leibniz rule, or product rule:

[(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x).]

Note the order of the factors; when R is not commutative this is important.

These two properties make D a derivation on A (see also module of relative differential forms for a discussion of a generalization).

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; namely, for every polynomial f(x) and every element r of R, there exists a nonnegative integer m_r and a polynomial g(x) such that

[f(x) = (x - r)^ g(x)]

where g(r) is not equal to 0. m_r is the multiplicity of r as a root of f. It follows from the Leibniz rule that in this situation, m_r is also the number of differentiations that must be performed on f(x) before r is not 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

[f(x)\,=\,x^6 + 1]

has no roots in R; however, its formal derivative is zero 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.

Formal derivative

See also