A real number x ∈ R is called algebraic if there exist integersa0, a1, a2, . . . , an ∈ Z, not all zero, such that anxn + an−1xn−1 + · · · + a1x + a0 = 0. (b) Fix n ∈ N, and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact thatevery polynomial has a finite number of roots, show that An is countable.

A real number x ∈ R is called algebraic if there exist integersa0, a1, a2, . . . , an ∈ Z, not all zero, such that anxn + an−1xn−1 + · · · + a1x + a0 = 0. (b) Fix n ∈ N, and let An be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact thatevery polynomial has a finite number of roots, show that An is countable.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 1E: 1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic...

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