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all algebraic numbers in a sequence∗
pahio†
2013-03-21 19:07:57
The beginning of the sequence of all algebraic numbers ordered as explained
in the parent entry is as follows:
√
√
√
√
√
3 −1+i 3
0; −1, 1; −2, − 12 , −i, i, 21 , 2; −3, −1−2 5 , − 2, − √12 , 1−2 5 , −1−i
,
, − 31 ,
2
2
√
√
√
√
√
√
√
−i 2, − √i2 , √i2 , i 2, 13 , 1−i2 3 , 1+i2 3 , −1+2 5 , √12 , 2, 1+2 5 , 3; . . .
The first number corresponds to the algebraic equation x = 0, the two
following numbers to the equations x ± 1 = 0, the six following to the equations
x ± 2 = 0, 2x ± 1 = 0, x2 + 1 = 0, the twenty following to the equations
x ± 3 = 0, 3x ± 1 = 0, x2 ± x ± 1 = 0, x2 ± 2 = 0, 2x2 ± 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!
∗ hAllAlgebraicNumbersInASequencei created: h2013-03-21i by: hpahioi version: h37003i
Privacy setting: h1i hResulti h11R04i h03E10i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
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