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!