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:
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“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)
“A child is born with the potential ability to learn Chinese or Swahili, play a kazoo, climb a tree, make a strudel or a birdhouse, take pleasure in finding the coordinates of a star. Genetic inheritance determines a child’s abilities and weaknesses. But those who raise a child call forth from that matrix the traits and talents they consider important.”
—Emilie Buchwald (20th century)
“A stated truth loses its grace, but a repeated error appears insipid and ridiculous.”
—Johann Wolfgang Von Goethe (1749–1832)
“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)