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'''Algebraic numbers''', are real numbers which are roots of some finite degree [[polynomial]] with integer coefficients.  Examples of algebraic numbers are 2, 1/3, and <math>\sqrt{2}</math>.

'''Algebraic numbers''', are numbers which are roots of some finite degree [[polynomial]] with integer coefficients.  Examples of real algebraic numbers are 2, 1/3, and <math>\sqrt{2}</math>.

In contrast, [[transcendental numbers]] are real numbers which are not roots of any finite degree polynomial with integer coefficients.  The most well-known transcendental numbers are [[e]] and <math>\pi</math> ([[pi]]), though this must be proven using advanced [[calculus]] or [[number theory]].

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 <math>\pi</math> ([[pi]]), though this 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]] because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

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]] because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

The algebraic numbers are closed under [[addition]], [[subtraction]], [[multiplication]], [[division]], [[square root]]s, [[radical]]s, and [[exponentiation]]. These properties imply that the algebraic numbers are an [[algebraically closed]] [[field]].

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 <math>\mathbb{A}</math> or <math>\bar{\mathbb{Q}}</math>.

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.  

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.  

A [[random]] real number is probably transcendental, though the exact value of this [[probability]] has baffled [[statistician]]s for almost 100 years.

[[Almost all]] real numbers are transcendental.

==Algebraic integers==

==Algebraic integers==

An algebraic integer is an algebraic number which is also an [[integer]], occurring in a subring of the algebraic number field.

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.

[[Category:Mathematics]]

[[Category:Mathematics]]