Can you answer only part 4 and part 5 The following problem is designed for you to prove that the set of all (real) algebraic numbers is denumerable. Anumber x* E R is an algebraic number if it is a root of a polynomial with integer coefficients, that is p(x*)= 0wherep(x) = anx" + an-1x"- + ..+ a1x + ao,for x E Rfor some a; E Z for i = 1, 2, . .. , n, and some n E N.i. Show that if A is a denumerable set then A x Z is denumerable as well (Hint: you may find simpler toprove that A × N is denumerable first).ii. Use i. to show that forn € N, the setZ x Z x . .. x Z,n timesis denumerable (Hint: try by induction).iii. Use ii. to prove that for each n E N the set Pn, defined asPn :={All polynomials with integer coefficients of degree

Answered: Image /qna-images/answer/59804347-eabe-4e9a-9458-dc31a9449076.jpg

iv. Use iii. to prove that the set P of all polynomials with integer coefficients is denumerable (Hint: use that P = UPn and Theorem 1.3.12). v. Use iv. and the fact that a polynomial of degree n has at most n real roots to show that the set of algebraic numbers is denumerable.

iv. Use iii. to prove that the set P of all polynomials with integer coefficients is denumerable (Hint: use that P = UPn and Theorem 1.3.12). v. Use iv. and the fact that a polynomial of degree n has at most n real roots to show that the set of algebraic numbers is denumerable.

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 22E

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Question

Can you answer only part 4 and part 5

The following problem is designed for you to prove that the set of all (real) algebraic numbers is denumerable. A
number x* E R is an algebraic number if it is a root of a polynomial with integer coefficients, that is p(x*)= 0
where
p(x) = anx" + an-1x"- + ..+ a1x + ao,
for x E R
for some a; E Z for i = 1, 2, . .. , n, and some n E N.
i. Show that if A is a denumerable set then A x Z is denumerable as well (Hint: you may find simpler to
prove that A × N is denumerable first).
ii. Use i. to show that forn € N, the set
Z x Z x . .. x Z,
n times
is denumerable (Hint: try by induction).
iii. Use ii. to prove that for each n E N the set Pn, defined as
Pn :=
{All polynomials with integer coefficients of degree <n },
is denumerable.
iv. Use iii. to prove that the set P of all polynomials with integer coefficients is denumerable (Hint: use that
P = U Pn and Theorem 1.3.12).
v. Use iv. and the fact that a polynomial of degree n has at most n real roots to show that the set of algebraic
numbers is denumerable.

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