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[parent] all algebraic numbers in a sequence (Result)

The beginning of the sequence of all algebraic numbers ordered as explained in the parent entry is as follows:

$ 0;\,\,-1,\,1;\,\,-2,\,-\frac{1}{2},\,-i,\,i,\,\frac{1}{2},\,2;\,\,-3,\, \frac{... ...qrt{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}{... ...qrt{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; instead we can list the equations satisfied by the numbers as far we want and tell how many roots they have. In principle, the number sequence does exist!



"all algebraic numbers in a sequence" is owned by pahio.
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Other names:  counting the algebraic numbers

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Cross-references: degree, equations, algebraic equation, number, algebraic numbers, sequence
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This is version 8 of all algebraic numbers in a sequence, born on 2005-05-03, modified 2006-10-17.
Object id is 7003, canonical name is AllAlgebraicNumbersInASequence.
Accessed 1720 times total.

Classification:
AMS MSC: 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
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