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)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)
“The great problem of American life [is] the riddle of authority: the difficulty of finding a way, within a liberal and individualistic social order, of living in harmonious and consecrated submission to something larger than oneself.... A yearning for self-transcendence and submission to authority [is] as deeply rooted as the lure of individual liberation.”
—Wilfred M. McClay, educator, author. The Masterless: Self and Society in Modern America, p. 4, University of North Carolina Press (1994)
“It is commonly said ... that ridicule is the best test of truth; for that it will not stick where it is not just. I deny it. A truth learned in a certain light, and attacked in certain words, by men of wit and humour, may, and often doth, become ridiculous, at least so far, that the truth is only remembered and repeated for the sake of the ridicule.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“The goal of every culture is to decay through over-civilization; the factors of decadence,—luxury, scepticism, weariness and superstition,—are constant. The civilization of one epoch becomes the manure of the next.”
—Cyril Connolly (1903–1974)