Editing Algebraic integer (section)

==Additional facts==
* Any number constructible out of the integers with roots, addition, and multiplication is an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible [[quintic]]s are not. This is the [[Abel–Ruffini theorem]]. <!-- what is the meaning of "most" roots of irreducible quintics? By counting, there are as many non-solvable as solvable quintics. Are coefficients of the quintic taken "randomly" from the integers? There ain't no such "random" integer! //--><!--How about this: Consider irreducible quintics of degree n, with integer coefficients with absolute value <= a. Does the proportion of them that are solvable not approach 0 as n and a go to infinity, whether separately or together?-->
* The ring of algebraic integers is a [[Bézout domain]], as a consequence of the [[principal ideal theorem]].
* If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the [[multiplicative inverse|reciprocal]] of that algebraic integer is also an algebraic integer, and each is a [[unit (ring theory)|unit]], an element of the [[group of units]] of the ring of algebraic integers.
* If {{math|''x''}} is an algebraic number then {{math|''a''<sub>''n''</sub>''x''}} is an algebraic integer, where {{mvar|x}} satisfies a polynomial {{math|''p''(''x'')}} with integer coefficients and where {{math|''a''<sub>''n''</sub>''x''<sup>''n''</sup>}} is the highest-degree term of {{math|''p''(''x'')}}.  The value {{math|1=''y'' = ''a''<sub>''n''</sub>''x''}} is an algebraic integer because it is a root of {{math|1=''q''(''y'') = ''a''{{su|b=''n''|p=''n''&nbsp;−&thinsp;1}}&thinsp;''p''(''y''{{hairsp}}/''a''<sub>''n''</sub>)}}, where {{math|''q''(''y'')}} is a monic polynomial with integer coefficients.
* If {{math|''x''}} is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer.  In fact, the denominator can always be chosen to be a positive integer.   The ratio is {{math|{{abs|''a''<sub>''n''</sub>}}''x'' / {{abs|''a''<sub>''n''</sub>}}}}, where {{mvar|x}} satisfies a polynomial {{math|''p''(''x'')}} with integer coefficients and where {{math|''a''<sub>''n''</sub>''x''<sup>''n''</sup>}} is the highest-degree term of {{math|''p''(''x'')}}.
* The only rational algebraic integers are the integers. Thus, if {{math|&alpha;}} is an algebraic integers and <math>\alpha\in\Q</math>, then <math>\alpha\in\Z</math>. This is a direct result of the [[rational root theorem]] for the case of a monic polynomial.