Formal derivative

The formal derivation is a term from the mathematical field of algebra. Through them, the derivative term from calculus is transmitted for functions on polynomials.

Since over a ring is not a number "between" two numbers exist, so there is no limit concept, the difference quotient can not be meaningfully defined and therefore there is no dissipation in the real sense. In order to use the concept of derivative nevertheless, this is formally defined for polynomials that the control and power factor control have been fulfilled.

Definition

Be a ring and denote the polynomial ring over in one indeterminate. For a polynomial

Is defined as the formal derivative

Properties

• For the formal derivation of the well-known calculation rules of differential calculus apply. In particular

• If there is in linear factors, ie, the zeros of are, then for the derivation

Application

If a body is, so is a Euclidean ring (especially factorial ), taking as Euclidean norm is used, if the coefficients of designated. The zeros of the gcd of and are just showing the multiple zeros of a decreased by 1 order, as the following calculation:

Be a multiple zero of, then apply with a polynomial and a. It follows, therefore.