Algebraic and Transcendental Numbers

[6/50%]

Show

LaTeX

Show LaTeX

Problem 1.2.2 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.2 "Basic Topology"

A complex number $z$ is said to be algebraic if there are integers $a_{0}, \ldots , a_{n}$ , not all zero, such that

a_{0} z^{n}+a_{1} z^{n-1}+\cdots +a_{n-1} z+a_{n} = 0

Prove that the set of all algebraic numbers is countable. Hint: For every positive integer $N$ there are only finitely many equations with

n+\left|a_{0}\right|+\left|a_{1}\right|+\cdots +\left|a_{n}\right| = N

Problem 1.2.3 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.2 "Basic Topology"

Prove that there exist real numbers which are not algebraic.

Problem 17.34 from Gallian J.A. "Contemporary Abstract Algebra" [Gal21]More problems in Chapter 17 "Factorization of Polynomials"

Given that $\pi$ is not the zero of a nonzero polynomial with rational coefficients, prove that $\pi^{2}$ cannot be written in the form $a\pi +b$ , where $a$ and $b$ are rational.

Problem 23 from Дехтярь М.И., Дудаков С.М., Карлов М.И., Карлов Б.Н. "Лекции по дискретной математике" [ДДК+21]More problems in Chapter 1 "Sets and Relations"

Prove that the set of algebraic numbers is countable.

Problem I.4.19 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.4 "Cardinal Numbers"

Prove the countability of the set of algebraic numbers, i.e., numbers that are roots of polynomials in one variable with integer coefficients.

Problem I.4.31 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.4 "Cardinal Numbers"

Prove the existence of transcendental (nonalgebraic) numbers.

References

[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.

[Gal21] Gallian J.A. Contemporary Abstract Algebra. 10th ed., CRC Press, 2021.

[ДДК+21] Дехтярь М.И., Дудаков С.М., Карлов М.И., Карлов Б.Н. Лекции по дискретной математике. 3-е изд., Тверской государственный университет, 2021.

[LaMa02] Lavrov I.A., Maksimova L.L. Задачи по теории множеств, математической логике и теории алгоритмов. 5-е изд., Физматлит, 2002.