Editing Algebraic number (section)

==<span class="anchor" id="Degree of an algebraic number"></span> Properties==
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[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4). The larger points come from polynomials with smaller integer coefficients.]]
*If a polynomial with rational coefficients is multiplied through by the [[least common denominator]], the resulting polynomial with integer coefficients has the same roots.  This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
*Given an algebraic number, there is a unique [[monic polynomial]] with rational coefficients of least [[degree of a polynomial|degree]] that has the number as a root. This polynomial is called its [[minimal polynomial (field theory)|minimal polynomial]]. If its minimal polynomial has degree {{mvar|n}}, then the algebraic number is said to be of '''degree {{mvar|n}}'''. For example, all [[rational number]]s have degree 1, and an algebraic number of degree 2 is a [[quadratic irrational]].
*The algebraic numbers are [[dense set|dense]] [[densely ordered|in the reals]].  This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
*The set of algebraic numbers is countable,{{sfn|Hardy|Wright|1972|p=160|loc=2008:205}}{{sfn|Niven|1956|loc=Theorem 7.5.}} and therefore its [[Lebesgue measure]] as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, [[Almost everywhere|"almost all"]] real and complex numbers are transcendental.
*All algebraic numbers are [[computable number|computable]] and therefore [[definable number|definable]] and [[arithmetical numbers|arithmetical]].
*For real numbers {{math|''a''}} and {{math|''b''}}, the complex number {{math|''a'' + ''bi''}} is algebraic if and only if both {{math|''a''}} and {{math|''b''}} are algebraic.{{sfn|Niven|1956|loc=Corollary 7.3.}}

===Degree of simple extensions of the rationals as a criterion to algebraicity===
For any {{math|&alpha;}}, the [[simple extension]] of the rationals by {{math|&alpha;}}, denoted by <math>\Q(\alpha) \equiv \{\sum_{i=-{n_1}}^{n_2} \alpha^i q_i | q_i\in \Q, n_1,n_2\in \N\}</math>, is of finite [[Degree of a field extension|degree]] if and only if {{math|&alpha;}} is an algebraic number.

The condition of finite degree means that there is a finite set <math>\{a_i | 1\le i\le k\}</math> in <math>\Q(\alpha)</math> such that <math>\Q(\alpha) = \sum_{i=1}^k a_i \Q</math>; that is, every member in <math>\Q(\alpha)</math> can be written as <math>\sum_{i=1}^k a_i q_i</math> for some rational numbers <math>\{q_i | 1\le i\le k\}</math> (note that the set <math>\{a_i\}</math> is fixed).

Indeed, since the <math>a_i-s</math> are themselves members of <math>\Q(\alpha)</math>, each can be expressed as sums of products of rational numbers and powers of {{math|&alpha;}}, and therefore this condition is equivalent to the requirement that for some finite <math>n</math>, <math>\Q(\alpha) = \{\sum_{i=-n}^n \alpha^{i} q_i | q_i\in \Q\}</math>.

The latter condition is equivalent to <math>\alpha^{n+1}</math>, itself a member of <math>\Q(\alpha)</math>, being expressible as <math>\sum_{i=-n}^n \alpha^i q_i</math> for some rationals <math>\{q_i\}</math>, so <math>\alpha^{2n+1} = \sum_{i=0}^{2n} \alpha^i q_{i-n}</math> or, equivalently, {{math|&alpha;}} is a root of <math>x^{2n+1}-\sum_{i=0}^{2n} x^i q_{i-n}</math>; that is, an algebraic number with a minimal polynomial of degree not larger than <math>2n+1</math>.

It can similarly be proven that for any finite set of algebraic numbers <math>\alpha_1</math>, <math>\alpha_2</math>... <math>\alpha_n</math>, the field extension <math>\Q(\alpha_1, \alpha_2, ... \alpha_n)</math> has a finite degree.