Editing Algebraic number (section)

==Algebraic integers==
[[Image:Leadingcoeff.png|thumb|Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top.]]
{{Main|Algebraic integer}}
An ''algebraic integer'' is an algebraic number that is a root of  a polynomial with integer coefficients with leading coefficient 1 (a [[monic polynomial]]). Examples of algebraic integers are <math>5 + 13 \sqrt{2},</math> <math>2 - 6i,</math> and <math display=inline>\frac{1}{2}(1+i\sqrt{3}).</math> Therefore, the algebraic integers constitute a proper [[superset]] of the [[integer]]s, as the latter are the roots of monic polynomials {{math|''x'' − ''k''}} for all <math>k \in \mathbb{Z}</math>. In this sense, algebraic integers are to algebraic numbers what [[integer]]s are to [[rational number]]s.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a [[Ring (mathematics)|ring]]. The name ''algebraic integer'' comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any [[algebraic number field|number field]] are in many ways analogous to the integers. If {{math|''K''}} is a number field, its [[ring of integers]] is the subring of algebraic integers in {{math|''K''}}, and is frequently denoted as {{math|''O<sub>K</sub>''}}. These are the prototypical examples of [[Dedekind domain]]s.