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:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“Great abilites are not requisite for an Historian; for in historical composition, all the greatest powers of the human mind are quiescent. He has facts ready to his hand; so there is no exercise of invention. Imagination is not required in any degree; only about as much as is used in the lowest kinds of poetry. Some penetration, accuracy, and colouring, will fit a man for the task, if he can give the application which is necessary.”
—Samuel Johnson (1709–1784)
“If everybody is looking for it, then nobody is finding it. If we were cultured, we would not be conscious of lacking culture. We would regard it as something natural and would not make so much fuss about it. And if we knew the real value of this word we would be cultured enough not to give it so much importance.”
—Pablo Picasso (1881–1973)
“Nothing in the nature around us is evil. This needs to be repeated since one of the human ways of talking oneself into inhuman acts is to cite the supposed cruelty of nature.”
—John Berger (b. 1926)
“Girls tend to attribute their failures to factors such as lack of ability, while boys tend to attribute failure to specific factors, including teachers’ attitudes. Moreover, girls avoid situations in which failure is likely, whereas boys approach such situations as a challenge, indicating that failure differentially affects self-esteem.”
—Michael Lewis (late–20th-century)