all algebraic numbers in a sequence

The beginning of the sequence of all algebraic numbers ordered as explained in the parent (http://planetmath.org/AlgebraicNumbersAreCountable) entry is as follows:

$0;-1,\mathrm{\hspace{0.17em}1};-2,-\frac{1}{2},-i,i,\frac{1}{2},\mathrm{\hspace{0.17em}2};-3,\frac{-1-\sqrt{5}}{2},-\sqrt{2},-\frac{1}{\sqrt{2}},\frac{1-\sqrt{5}}{2},\frac{-1-i\sqrt{3}}{2},\frac{-1+i\sqrt{3}}{2},-\frac{1}{3},$

$-i\sqrt{2},-\frac{i}{\sqrt{2}},\frac{i}{\sqrt{2}},i\sqrt{2},\frac{1}{3},\frac{1-i\sqrt{3}}{2},\frac{1+i\sqrt{3}}{2},\frac{-1+\sqrt{5}}{2},\frac{1}{\sqrt{2}},\sqrt{2},\frac{1+\sqrt{5}}{2},\mathrm{\hspace{0.17em}3};\mathrm{\dots }$

The first number corresponds to the algebraic equation  $x=0$,  the two following numbers to the equations  $x\pm 1=0$,  the six following to the equations  $x\pm 2=0$,  $2x\pm 1=0$,  $x^2+1=0$,  the twenty following to the equations  $x\pm 3=0$,  $3x\pm 1=0$,  $x^2\pm x\pm 1=0$,  $x^2\pm 2=0$,  $2x^2\pm 1=0$.

In practice, one cannot continue the sequence very far since the higher degree equations – quintic and so on – are non-solvable by radicals (http://planetmath.org/NthRoot); instead we can list the equations satisfied by the numbers as far we want and tell how many roots (http://planetmath.org/Root) they have.  In principle, the number sequence does exist!

Title	all algebraic numbers in a sequence
Canonical name	AllAlgebraicNumbersInASequence
Date of creation	2013-03-22 15:13:58
Last modified on	2013-03-22 15:13:58
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Result
Classification	msc 11R04
Classification	msc 03E10
Synonym	counting the algebraic numbers
Related topic	FieldOfAlgebraicNumbers