Application To Finding Repeated Factors
As in calculus, the derivative detects multiple roots: if R is a field then R 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 mr and a polynomial g(x) such that
where g(r) is not equal to 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 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 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 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.
Read more about this topic: Formal Derivative
Famous quotes containing the words application to, application, finding, repeated and/or factors:
“Preaching is the expression of the moral sentiment in application to the duties of life.”
—Ralph Waldo Emerson (1803–1882)
“I conceive that the leading characteristic of the nineteenth century has been the rapid growth of the scientific spirit, the consequent application of scientific methods of investigation to all the problems with which the human mind is occupied, and the correlative rejection of traditional beliefs which have proved their incompetence to bear such investigation.”
—Thomas Henry Huxley (1825–95)
“For anyone addicted to reading commonplace books ... finding a good new one is much like enduring a familiar recurrence of malaria, with fever, fits of shaking, strange dreams. Unlike a truly paludismic ordeal, however, the symptoms felt while savoring a collection of one man’s pet quotations are voluptuously enjoyable ...”
—M.F.K. Fisher (1908–1992)
“Manners are the happy way of doing things; each once a stroke of genius or of love—now repeated and hardened into usage. They form at last a rich varnish, with which the routine of life is washed, and its details adorned. If they are superficial, so are the dewdrops which give such depth to the morning meadows.”
—Ralph Waldo Emerson (1803–1882)
“Language makes it possible for a child to incorporate his parents’ verbal prohibitions, to make them part of himself....We don’t speak of a conscience yet in the child who is just acquiring language, but we can see very clearly how language plays an indispensable role in the formation of conscience. In fact, the moral achievement of man, the whole complex of factors that go into the organization of conscience is very largely based upon language.”
—Selma H. Fraiberg (20th century)