Editing Algebraic number (section)

==Examples==
* All [[rational number]]s are algebraic. Any rational number, expressed as the quotient of an [[integer]] {{mvar|a}} and a (non-zero) [[natural number]] {{mvar|b}}, satisfies the above definition, because {{math|''x'' {{=}} {{sfrac|''a''|''b''}}}} is the root of a non-zero polynomial, namely {{math|''bx'' − ''a''}}.<ref>Some of the following examples come from {{harvtxt|Hardy|Wright|1972|pp=159–160, 178–179}}</ref>
* [[Quadratic irrational number]]s, irrational solutions of a quadratic polynomial {{math|''ax''{{sup|2}} + ''bx'' + ''c''}} with integer coefficients {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}, are algebraic numbers. If the quadratic polynomial is monic ({{math|''a'' {{=}} 1}}), the roots are further qualified as [[quadratic integer]]s.
** [[Gaussian integer]]s, complex numbers {{math|''a'' + ''bi''}} for which both {{mvar|a}} and {{mvar|b}} are integers, are also quadratic integers. This is because {{math|''a'' + ''bi''}} and {{math|''a'' − ''bi''}} are the two roots of the quadratic {{math|''x''{{sup|2}} − 2''ax'' + ''a''{{sup|2}} + ''b''{{sup|2}}}}.
* A [[constructible number]] can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the [[Arithmetic operations|basic arithmetic operations]] and the extraction of square roots. (By designating cardinal directions for +1, −1, +{{mvar|i}}, and −{{mvar|i}}, complex numbers such as <math>3+i \sqrt{2}</math> are considered constructible.)
* Any expression formed from algebraic numbers using any finite combination of the basic arithmetic operations and extraction of [[nth root|{{mvar|n}}th roots]] gives another algebraic number.
* Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of {{mvar|n}}th roots (such as the roots of {{math|''x''<sup>5</sup> − ''x'' + 1}}). [[Abel–Ruffini theorem|That happens with many]] but not all polynomials of degree 5 or higher.
* Values of [[trigonometric functions]] of rational multiples of {{pi}} (except when undefined): for example, {{math|cos {{sfrac|{{math|π}}|7}}}}, {{math|cos {{sfrac|3{{math|π}}|7}}}}, and {{math|cos {{sfrac|5{{math|π}}|7}}}} satisfy {{math|8''x''<sup>3</sup> − 4''x''<sup>2</sup> − 4''x'' + 1 {{=}} 0}}. This polynomial is [[irreducible polynomial|irreducible]] over the rationals and so the three cosines are ''conjugate'' algebraic numbers. Likewise, {{math|tan {{sfrac|3{{math|π}}|16}}}}, {{math|tan {{sfrac|7{{math|π}}|16}}}}, {{math|tan {{sfrac|11{{math|π}}|16}}}}, and {{math|tan {{sfrac|15{{math|π}}|16}}}} satisfy the irreducible polynomial {{math|''x''<sup>4</sup> − 4''x''<sup>3</sup> − 6''x''<sup>2</sup> + 4''x'' + 1 {{=}} 0}}, and so are conjugate [[algebraic integer]]s. This is the equivalent of angles which, when measured in degrees, have rational numbers.{{sfn|Garibaldi|2008}}
* Some but not all irrational numbers are algebraic:
** The numbers <math>\sqrt{2}</math> and <math>\frac{ \sqrt[3]{3} }{ 2 }</math> are algebraic since they are roots of polynomials {{math|''x''<sup>2</sup> − 2}} and {{math|8''x''<sup>3</sup> − 3}}, respectively.
** The [[golden ratio]] {{mvar|φ}} is algebraic since it is a root of the polynomial {{math|''x''<sup>2</sup> − ''x'' − 1}}.
** The numbers [[pi|{{pi}}]] and [[e (mathematical constant)|e]] are not algebraic numbers (see the [[Lindemann–Weierstrass theorem]]).<ref>Also, [[Liouville number|Liouville's theorem]] can be used to "produce as many examples of transcendental numbers as we please," cf. {{harvtxt|Hardy|Wright|1972|p=161ff}}</ref>