Algebraic numbers

Algebraic numbers are those complex numbers which are roots of some polynomial with integer coefficients. Examples of real algebraic numbers are 2, 1/3, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} , which are roots of the polynomials $x-2=0$, $3x-1=0$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-2=0} , respectively.

In contrast, transcendental numbers are numbers which are not roots of any finite degree polynomial with integer coefficients. The most well-known transcendental numbers are e and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} (pi), though the fact that these are transcendental must be proven using advanced calculus or number theory.

The set of algebraic numbers is countably infinite. This follows from the fact that the set of finite degree polynomials with integer coefficients is countably infinite. Conversely, the set of transcendental numbers is uncountable, as this set is defined as the complex numbers (which are uncountable), minus a countable set.

The algebraic numbers are closed under addition, subtraction, multiplication and division. These properties show that the algebraic numbers are a field. In addition every root of a polynomial of finite degree with algebraic coefficients is an algebraic number, so it is an algebraically closed field. The field of algebraic numbers is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\mathbb{Q}}} .

If a is an algebraic number and t is a transcendental number, then a+t, a-t, at, a/t, a^t, and t^a are all transcendental.

Almost all real numbers are transcendental.

Algebraic integers

An algebraic integer is an algebraic number which is a root of a monic polynomial with integer coefficients. The algebraic integers form a subring of the algebraic numbers. However, they do not form a subfield: for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} is a root of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-3=0} but its multiplicative inverse Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/3} is not a root of any monic polynomial.