Editing Algebraic number (section)

===Algebraic closure===
Every root of a polynomial equation whose coefficients are ''algebraic numbers'' is again algebraic. That can be rephrased by saying that the field of algebraic numbers is [[algebraically closed field|algebraically closed]]. In fact, it is the smallest algebraically closed field containing the rationals and so it is called the [[algebraic closure]] of the rationals.

That the field of algebraic numbers is algebraically closed can be proven as follows: Let {{math|&beta;}} be a root of a polynomial <math> \alpha_0 + \alpha_1 x + \alpha_2 x^2 ... +\alpha_n x^n</math> with coefficients that are algebraic numbers <math>\alpha_0</math>, <math>\alpha_1</math>, <math>\alpha_2</math>... <math>\alpha_n</math>. The field extension <math>\Q^\prime \equiv \Q(\alpha_1, \alpha_2, ... \alpha_n)</math> then has a finite degree with respect to <math>\Q</math>. The simple extension <math>\Q^\prime(\beta)</math> then has a finite degree with respect to <math>\Q^\prime</math> (since all powers of {{math|&beta;}} can be expressed by powers of up to <math>\beta^{n-1}</math>). Therefore, <math>\Q^\prime(\beta) = \Q(\beta, \alpha_1, \alpha_2, ... \alpha_n)</math> also has a finite degree with respect to <math>\Q</math>. Since <math>\Q(\beta)</math> is a linear subspace of <math>\Q^\prime(\beta)</math>, it must also have a finite degree with respect to <math>\Q</math>, so {{math|&beta;}} must be an algebraic number.