2.5 notes
... • Given a polynomial, the number of positive real zeros is either equal to the number of sign changes in the polynomial, or less than that by a factor of 2. • The number of negative real zeros is equal to the number of sign changes in f(-x) or or less than that by a factor of 2 ...
... • Given a polynomial, the number of positive real zeros is either equal to the number of sign changes in the polynomial, or less than that by a factor of 2. • The number of negative real zeros is equal to the number of sign changes in f(-x) or or less than that by a factor of 2 ...
Math 322, Fall Term 2011 Final Exam
... (a) Show that f (x) = x3 + 2x + 2 is irreducible in F3 [x] (F3 is the finite field with three elements) and use this fact to construct a field with 27 elements that contains F3 . (b) Consider the polynomial f (x) = (x2 + 1)(x2 − 2) over Q. Find a field extension of Q where f (x) splits completely in ...
... (a) Show that f (x) = x3 + 2x + 2 is irreducible in F3 [x] (F3 is the finite field with three elements) and use this fact to construct a field with 27 elements that contains F3 . (b) Consider the polynomial f (x) = (x2 + 1)(x2 − 2) over Q. Find a field extension of Q where f (x) splits completely in ...
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... Attempt all questions. All your answers, including yes/no answers, must be supported by proofs, unless indicated otherwise. As usual, IR denotes the real numbers, e the complex numbers, Q the r;1tional numbers, Z the integers,,, and Zn the integers modulo n. If you use any significant theorems in yo ...
... Attempt all questions. All your answers, including yes/no answers, must be supported by proofs, unless indicated otherwise. As usual, IR denotes the real numbers, e the complex numbers, Q the r;1tional numbers, Z the integers,,, and Zn the integers modulo n. If you use any significant theorems in yo ...
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MATH 521A: Abstract Algebra Homework 7 Solutions 1. Consider
... different ones, such as f (x) = (x + 2)(x + x + 1), and 10 ways of picking the square of one, such as f (x) = (x2 + 2)2 . Hence there are 45 + 10 = 55 answers to this question. 6. Factor x7 − x as a product of irreducibles in Z7 [x]. By Fermat’s Little Theorem, x7 ≡ x (mod 7), for all integer x. Hen ...
... different ones, such as f (x) = (x + 2)(x + x + 1), and 10 ways of picking the square of one, such as f (x) = (x2 + 2)2 . Hence there are 45 + 10 = 55 answers to this question. 6. Factor x7 − x as a product of irreducibles in Z7 [x]. By Fermat’s Little Theorem, x7 ≡ x (mod 7), for all integer x. Hen ...
PDF
... Elements of R[X] are called polynomials in the indeterminate X with coefficients in R. The ring elements a0 , . . . , aN are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number N for which aN 6= 0, if such an N exists. When a polynomial has all of its ...
... Elements of R[X] are called polynomials in the indeterminate X with coefficients in R. The ring elements a0 , . . . , aN are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number N for which aN 6= 0, if such an N exists. When a polynomial has all of its ...
