By S. E. Payne
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This energetic creation to degree concept and Lebesgue integration is encouraged through the ancient questions that ended in its improvement. the writer stresses the unique objective of the definitions and theorems, highlighting the problems mathematicians encountered as those rules have been sophisticated. the tale starts off with Riemann's definition of the necessary, after which follows the efforts of these who wrestled with the problems inherent in it, till Lebesgue ultimately broke with Riemann's definition.
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Pi = ti − tj j=i Then each pi has degree n, so belongs to V , and Ej (pi ) = pi (tj ) = δij . 5) It will turn out that B = (p0 , . . , pn ) is a basis for V , and then Eq. 5 expresses what we mean by saying that B ∗ is the basis dual to B. If f = n i=0 ci pi , then for each j f (tj ) = ci pi (tj ) = cj . 6) i So if f is the zero polynomial, each cj must equal 0, implying that the list (p0 , . . , pn ) is linearly independent in V . Since (1, x, x2 , . . , xn ) is a basis for V , clearly dim(V ) = n + 1.
Fk are relatively prime. i=j 60 CHAPTER 5. POLYNOMIALS 8. Using the same notation as in the preceding problem, suppose that f = p1 · · · pk is a product of distinct non-scalar irreducible polynomials over F . So fj = f /pj . Show that f = p 1 f 1 + p2 f 2 + · · · + pk f k . 9. Let f ∈ F [x] have derivative f . Then f is a product of distinct irreducible polynomials over F if and only if f and f are relatively prime. 10. Euclidean algorithm for polynomials Let f (x) and g(x) be polynomials over F for which deg(f (x)) ≥ deg(g(x)) ≥ 1.
7. Let M be any non-zero ideal in F [x]. Then there is a unique monic polynomial d ∈ F [x] such that M is the principal ideal dF [x] generated by d. Proof. Among all nonzero polynomials in M there is (at least) one of minimal degree. Hence there must be a monic polynomial d of least degree in M . Suppose that f is any element of M . We can divide f by d and get a unique quotient and remainder: f = qd + r where deg(r) < deg(d). Then r = f − qd ∈ M , but deg(r) is less than the smallest degree of any nonzero polynomial in M .