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'''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>, which are roots of the polynomials <math>x-2=0</math>, <math>3x-1=0</math>, <math>x^2-2=0</math>, respectively.

'''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 <math>\sqrt{2}</math>, which are roots of the polynomials <math>x-2=0</math>, <math>3x-1=0</math>, <math>x^2-2=0</math>, 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 <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 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]] 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]], 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_(mathematics)|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>.

The algebraic numbers are closed under [[addition]], [[subtraction]], [[multiplication]] and [[division]]. These properties show that the algebraic numbers are a [[Field_(mathematics)|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.  

[[Almost all]] real numbers are transcendental.

[[Measure theory|Almost all]] real numbers are transcendental.

==Algebraic integers==

==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.

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, <math>3</math> is a root of <math>x-3=0</math> but its multiplicative inverse <math>1/3</math> is not a root of any monic polynomial. 

[[Category:Number Theory]]

[[Category:Number Theory]]