Question

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

Short answer

Briefly explain how to prove that the set of all algebraic numbers is countable. To prove that the set of all algebraic numbers is countable, we first show that there are countably many polynomials with integer coefficients using a function that assigns each polynomial a unique natural number. Then, we recognize that each polynomial has finitely many roots, which are algebraic numbers. By listing the roots of all the countably many integer polynomials, we create a countable union of finite sets, which implies that there are countably many algebraic numbers. This demonstrates that there exists an injection from the set of algebraic numbers to the set of natural numbers, proving that the set of all algebraic numbers is countable.

Step-by-step solution

Analyzing polynomials with integer coefficients

For every positive integer \(N\), we are given that there are only finitely many equations with \(n+\left|a_{0}\right|+\left|a_{1}\right|+\cdots+\left|a_{n}\right|=N\). To show that there are countably many such polynomials, we can define a function that assigns each polynomial a unique natural number.

Constructing an injection from polynomials to natural numbers

Consider the function \(f(P) = n+\left|a_{0}\right|+\left|a_{1}\right|+\cdots+\left|a_{n}\right|\) for a polynomial \(P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}\). This function assigns a unique natural number to each integer polynomial \(P(x)\). Since there are finitely many polynomials for every positive integer \(N\), the set of integer polynomials is a countable union of finite sets, and is therefore countable.

Counting the algebraic numbers

Now, we need to count the algebraic numbers. Recall that an algebraic number \(z\) is a root of an integer polynomial. Since each polynomial has finitely many roots, we can create a list of all algebraic numbers by going through the list of integer polynomials and listing their roots. Since there are countably many integer polynomials, this will be a countable union of finite sets, implying that there are countably many algebraic numbers. In conclusion, we have shown that there exists an injection from the set of algebraic numbers to the set of natural numbers, which proves that the set of all algebraic numbers is countable.

Key concepts

Countability

Countability is a concept in mathematics that helps us understand the size of infinite sets. A set is considered countable if its elements can be paired one-to-one with natural numbers. This means there is a way to list the elements without missing any, just as you can list natural numbers: 1, 2, 3, and so on.

In the context of algebraic numbers, countability is important because it shows that even though there are infinitely many algebraic numbers, they are not 'too infinite' — they are still manageable in size, just like the set of natural numbers.

To prove that the set of all algebraic numbers is countable, mathematicians construct functions that pair each algebraic number with distinct natural numbers systematically. This ensures that every algebraic number is accounted for without leaving any out. By understanding countability, we see that even infinite sets can vary greatly in size and structure.

Polynomial Roots

Polynomial roots are values of the variable that turn the polynomial into zero. If you have a polynomial equation like \(a_{0}x^n + a_{1}x^{n-1} + \ldots + a_n = 0\), the values of \(x\) that satisfy this equation are the roots.

Understanding polynomial roots is crucial for solving polynomial equations. Algebraic numbers are defined as numbers that are roots of polynomials with integer coefficients. This means if \(z\) is an algebraic number, there exists a polynomial equation formed only with integers such that \(z\) makes the equation zero.

Integer Polynomials

Integer polynomials refer to polynomial expressions where the coefficients \(a_0, a_1, \ldots, a_n\) are integers. These polynomials take the form \(P(x) = a_{0}x^n + a_{1}x^{n-1} + \ldots + a_n\).

Working with integer polynomials is essential when discussing algebraic numbers since these numbers are, by definition, solutions to such polynomials.

• Integer polynomials have a finite degree and finite coefficients.
• For any positive integer \(N\), there are finite polynomial equations such that the sum of the degree and the absolute value of coefficients equals \(N\).

When considering algebraic numbers, each polynomial can have one or more roots. Since each integer polynomial has a finite degree which determines the maximum number of roots, listing these roots results in a manageable, countable set of solutions.

Complex Numbers

Complex numbers extend the range of algebraic numbers and consist of real and imaginary parts. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2=-1\).

Algebraic numbers often belong to the complex number system because complex numbers can be roots of polynomial equations. For instance, the polynomial \(x^2 + 1 = 0\) has roots \(i\) and \(-i\), which are complex numbers. Since they are roots of a polynomial with integer coefficients, they are also algebraic numbers.

Complex numbers thus play an essential role when dealing with algebraic numbers, especially since many polynomials do not restrict roots to only real numbers. This broadens the scope and allows us to find all possible roots.