Editing Algebraic integer (section)

==Examples==
* The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of <math>\mathbb{Q}</math> and {{mvar|A}} is exactly <math>\mathbb{Z}</math>. The rational number {{math|{{sfrac|''a''|''b''}}}} is not an algebraic integer unless {{mvar|b}} [[divisor|divides]] {{mvar|a}}. The leading coefficient of the polynomial {{math|''bx'' − ''a''}} is the integer {{mvar|b}}.
* The [[square root]] <math>\sqrt{n}</math> of a nonnegative integer {{mvar|n}} is an algebraic integer, but is [[irrational number|irrational]] unless {{mvar|n}} is a [[square number|perfect square]].
*If {{mvar|d}} is a [[square-free integer]] then the [[field extension|extension]] <math>K = \mathbb{Q}(\sqrt{d}\,)</math> is a [[quadratic field extension|quadratic field]] of rational numbers. The ring of algebraic integers {{math|{{mathcal|O}}<sub>''K''</sub>}} contains <math>\sqrt{d}</math> since this is a root of the monic polynomial {{math|''x''<sup>2</sup> − ''d''}}. Moreover, if {{math|''d'' ≡ 1 [[modular arithmetic|mod]] 4}}, then the element <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> is also an algebraic integer. It satisfies the polynomial {{math|''x''<sup>2</sup> − ''x'' + {{sfrac|1|4}}(1 − ''d'')}} where the [[constant term]] {{math|{{sfrac|1|4}}(1 − ''d'')}} is an integer. The full ring of integers is generated by <math>\sqrt{d}</math> or <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> respectively. See [[Quadratic integer]] for more.
*The ring of integers of the field <math>F = \Q[\alpha]</math>, {{math|1=''α'' = {{radic|''m''|3}}}}, has the following [[integral basis]], writing {{math|1=''m'' = ''hk''<sup>2</sup>}} for two [[square-free integer|square-free]] [[coprime]] integers {{mvar|h}} and {{mvar|k}}:<ref>{{cite book| last1=Marcus | first1=Daniel A. | title=Number Fields |edition=3rd | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90279-1 | year=1977 |at=ch. 2, p. 38 and ex. 41}}</ref> <math display="block">\begin{cases}
1, \alpha, \dfrac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \bmod 9 \\
1, \alpha, \dfrac{\alpha^2}k  & \text{otherwise}
\end{cases}</math>
* If {{mvar|ζ<sub>n</sub>}} is a [[primitive root of unity|primitive]] {{mvar|n}}th [[root of unity]], then the ring of integers of the [[cyclotomic field]] <math>\Q(\zeta_n)</math> is precisely <math>\Z[\zeta_n]</math>.
* If {{mvar|α}} is an algebraic integer then {{math|1=''β'' = {{radic|''α''|''n''}}}} is another algebraic integer. A polynomial for {{mvar|β}} is obtained by substituting {{math|''x<sup>n</sup>''}} in the polynomial for {{mvar|α}}.