Editing Algebraic number (section)

===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.