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Orderings of algebraic numbers
From OeisWiki
The algebraic numbers, roots of nonconstant polynomials with integer coefficients, may be sorted with a graded ordering, where we first order by increasing of degree plus sum of absolute values of coefficients for all minimal polynomials (first grade of the ordering), then by increasing degrees corresponding to that first grade (second grade of the ordering), then by increasing absolute values of coefficients of polynomials corresponding to that second grade (third grade of the ordering), (then same for absolute values of coefficients then same for absolute values of coefficients ), then for all the roots of the current polynomial of degree . Since the algebraic numbers may be ordered, we have a one-to-one and onto mapping from the natural numbers to the algebraic numbers, thus the algebraic numbers are countable (denumerable).
In the following table, is the degree of the polynomial. The polynomials which are not minimal polynomials are in red.
Polynomial
signature
| Polynomial
equation | Integer
coefficients polynomials factorization | Roots | |
---|---|---|---|---|
2 | (1; 1, 0) | 0 | ||
3 | (1; 1, -1) | 1 | ||
3 | (1; 1, +1) | -1 | ||
3 | (1; 2, 0) | ● | ||
3 | (2; 1, 0, 0) | ● | ||
4 | (1; 1, -2) | 2 | ||
4 | (1; 1, +2) | -2 | ||
4 | (1; 2, -1) | 1/2 | ||
4 | (1; 2, +1) | -1/2 | ||
4 | (1; 3, 0) | ● | ||
4 | (2; 1, 0, -1) | ● | ||
4 | (2; 1, 0, +1) | |||
4 | (2; 1, -1, 0) | ● | ||
4 | (2; 1, +1, 0) | ● | ||
4 | (3; 1, 0, 0, 0) | ● | ||
5 | (1; 1, -3) | 3 | ||
5 | (1; 1, +3) | -3 | ||
5 | (1; 2, -2) | ● | ||
5 | (1; 2, +2) | ● | ||
5 | (1; 3, -1) | 1/3 | ||
5 | (1; 3, +1) | -1/3 | ||
5 | (1; 4, 0) | ● | ||
5 | (2; 1, 0, -2) | |||
5 | (2; 1, 0, +2) | |||
5 | (2; 1, -1, -1) | |||
5 | (2; 1, -1, +1) | |||
5 | (2; 1, +1, -1) | |||
5 | (2; 1, +1, +1) | |||
5 | (2; 1, -2, 0) | ● | ||
5 | (2; 1, +2, 0) | ● | ||
5 | (2; 2, 0, -1) | |||
5 | (2; 2, 0, +1) | |||
5 | (2; 2, -1, 0) | ● | ||
5 | (2; 2, +1, 0) | ● | ||
5 | (2; 3, 0, 0) | ● | ||
5 | (3; 1, 0, 0, -1) | ● | ||
5 | (3; 1, 0, 0, +1) | ● | ||
5 | (3; 1, 0, -1, 0) | ● | ||
5 | (3; 1, 0, +1, 0) | ● | ||
5 | (3; 1, -1, 0, 0) | ● | ||
5 | (3; 1, +1, 0, 0) | ● | ||
5 | (3; 2, 0, 0, 0) | ● | ||
6 | (1; 1, -4) | 4 | ||
6 | (1; 1, +4) | -4 | ||
6 | (1; 2, -3) | 3/2 | ||
6 | (1; 2, +3) | -3/2 | ||
6 | (1; 3, -2) | 2/3 | ||
6 | (1; 3, +2) | -2/3 | ||
6 | (1; 4, -1) | 1/4 | ||
6 | (1; 4, +1) | -1/4 | ||
6 | (1; 5, 0) | ● | ||
6 | (2; 1, 0, -3) | |||
6 | (2; 1, 0, +3) | |||
6 | (2; 1, -1, -2) | ● | ||
6 | (2; 1, -1, +2) | |||
6 | (2; 1, +1, -2) | ● | ||
6 | (2; 1, +1, +2) | |||
6 | (2; 1, -2, -1) | |||
6 | (2; 1, -2, +1) | ● | ||
6 | (2; 1, +2, -1) | |||
6 | (2; 1, +2, +1) | ● | ||
6 | (2; 1, -3, 0) | ● | ||
6 | (2; 1, +3, 0) | ● | ||
6 | (2; 2, -1, -1) | ● | ||
6 | (2; 2, -1, +1) | |||
6 | (2; 2, +1, -1) | ● | ||
6 | (2; 2, +1, +1) | |||
6 | (2; 2, -2, 0) | ● | ||
6 | (2; 2, +2, 0) | ● | ||
6 | (2; 3, 0, -1) | |||
6 | (2; 3, 0, +1) | |||
6 | (2; 3, -1, 0) | ● | ||
6 | (2; 3, +1, 0) | ● | ||
6 | (2; 4, 0, 0) | ● | ||
6 | (3; 1, 0, 0, -2) | |||
6 | (3; 1, 0, 0, +2) | |||
6 | (3; 1, 0, -1, -1) | ● | ||
6 | (3; 1, 0, -1, +1) | ● | ||
6 | (3; 1, 0, +1, -1) | ???^{[1]} | ||
6 | (3; 1, 0, +1, +1) | ???^{[1]} | ||
6 | (3; 1, 0, -2, 0) | ● | ||
6 | (3; 1, 0, +2, 0) | ● | ||
6 | (3; 1, -1, 0, -1) | ???^{[1]} | ||
6 | (3; 1, -1, 0, +1) | ???^{[1]} | ||
6 | (3; 1, +1, 0, -1) | ???^{[1]} | ||
6 | (3; 1, +1, 0, +1) | ???^{[1]} | ||
6 | (3; 1, -1, -1, 0) | ● | ||
6 | (3; 1, -1, +1, 0) | ● | ||
6 | (3; 1, +1, -1, 0) | ● | ||
6 | (3; 1, +1, +1, 0) | ● | ||
6 | (3; 1, -2, 0, 0) | ● | ||
6 | (3; 1, +2, 0, 0) | ● | ||
6 | (3; 2, 0, 0, -1) | |||
6 | (3; 2, 0, 0, +1) | |||
6 | (3; 2, 0, -1, 0) | ● | ||
6 | (3; 2, 0, +1, 0) | ● | ||
6 | (3; 2, -1, 0, 0) | ● | ||
6 | (3; 2, +1, 0, 0) | ● |
The algebraic numbers can thus be enumerated as (Cf. A??????; note that algebraic number sequences are not in the OEIS!)
Graded orderings of rational numbers
The algebraic numbers of degree are the roots of the polynomial equation
with
This gives graded orderings of rational numbers.
See also
- Orderings of algebraic numbers and permutation of the algebraic numbers
- Orderings of rational numbers and permutation of the rational numbers
- Orderings of integers and permutation of the integers
- Orderings of positive integers and permutation of the positive integers
Notes
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} Complicated cubic roots obtainable by the cubic formula.