# 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:

$0;\,\,-1,\,1;\,\,-2,\,-\frac{1}{2},\,-i,\,i,\,\frac{1}{2},\,2;\,\,-3,\,\frac{-% 1-\sqrt{5}}{2},\,-\sqrt{2},\,-\frac{1}{\sqrt{2}},\,\frac{1-\sqrt{5}}{2},\,% \frac{-1-i\sqrt{3}}{2},\,\frac{-1+i\sqrt{3}}{2},\,-\frac{1}{3},\,$

$-i\sqrt{2},\,-\frac{i}{\sqrt{2}},\,\frac{i}{\sqrt{2}},\,i\sqrt{2},\,\frac{1}{3% },\,\frac{1-i\sqrt{3}}{2},\,\frac{1+i\sqrt{3}}{2},\,\frac{-1+\sqrt{5}}{2},\,% \frac{1}{\sqrt{2}},\,\sqrt{2},\,\frac{1+\sqrt{5}}{2},\,3;\,\ldots$

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 AllAlgebraicNumbersInASequence 2013-03-22 15:13:58 2013-03-22 15:13:58 pahio (2872) pahio (2872) 11 pahio (2872) Result msc 11R04 msc 03E10 counting the algebraic numbers FieldOfAlgebraicNumbers