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Orderings of algebraic numbers

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The algebraic numbers, roots of nonconstant polynomials with integer coefficients, may be sorted with a graded ordering, where we first order by increasing \scriptstyle d + \sum_{i=0}^{d} |a_{i}| \, of degree \scriptstyle d \, 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 \scriptstyle |a_{n}| \, of polynomials corresponding to that second grade (third grade of the ordering), (then same for absolute values of coefficients \scriptstyle |a_{n-1}|,\, \ldots,\, \, then same for absolute values of coefficients \scriptstyle |a_{0}| \,), then for all the \scriptstyle d \, roots of the current polynomial of degree \scriptstyle d \,. 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, \scriptstyle d \, is the degree of the polynomial. The polynomials which are not minimal polynomials are in red.

Graded lexicographic ordering of algebraic numbers
\scriptstyle d + \sum_{i=0}^{d} |a_i| \, Polynomial

signature

\scriptstyle (d; \, a_d, \, \ldots, \, a_0), \,

\scriptstyle d \,\ge\, 1,\, a_d \,\ge\, 1 \,

Polynomial

equation

Integer

coefficients

polynomials

factorization

Roots
2 (1; 1, 0) \scriptstyle x + 0 \,=\, 0 \, \scriptstyle x \,=\, 0 \, 0
3 (1; 1, -1) \scriptstyle x - 1 \,=\, 0 \, \scriptstyle x - 1 \,=\, 0 \, 1
3 (1; 1, +1) \scriptstyle x + 1 \,=\, 0 \, \scriptstyle x + 1 \,=\, 0 \, -1
3 (1; 2, 0) \scriptstyle 2 x + 0 \,=\, 0 \, \scriptstyle 2 x \,=\, 0 \,
3 (2; 1, 0, 0) \scriptstyle x^2 + 0 x + 0 \,=\, 0 \, \scriptstyle x^2 \,=\, 0 \,
4 (1; 1, -2) \scriptstyle x - 2 \,=\, 0 \, \scriptstyle x - 2 \,=\, 0 \, 2
4 (1; 1, +2) \scriptstyle x + 2 \,=\, 0 \, \scriptstyle x + 2 \,=\, 0 \, -2
4 (1; 2, -1) \scriptstyle 2 x - 1 \,=\, 0 \, \scriptstyle 2 x - 1 \,=\, 0 \, 1/2
4 (1; 2, +1) \scriptstyle 2 x + 1 \,=\, 0 \, \scriptstyle 2 x + 1 \,=\, 0 \, -1/2
4 (1; 3, 0) \scriptstyle 3 x + 0 \,=\, 0 \, \scriptstyle 3 x \,=\, 0 \,
4 (2; 1, 0, -1) \scriptstyle x^2 + 0 x - 1 \,=\, 0 \, \scriptstyle (x - 1) (x + 1) \,=\, 0 \,
4 (2; 1, 0, +1) \scriptstyle x^2 + 0 x + 1 \,=\, 0 \, \scriptstyle x^2 + 1 \,=\, 0 \, \scriptstyle (i,\, -i) \,=\, \big( e^{\frac{i \pi}{2}},\, e^{\frac{i 3 \pi}{2}} \big) \,
4 (2; 1, -1, 0) \scriptstyle x^2 - x + 0 \,=\, 0 \, \scriptstyle x (x - 1) \,=\, 0 \,
4 (2; 1, +1, 0) \scriptstyle x^2 + x + 0 \,=\, 0 \, \scriptstyle x (x + 1) \,=\, 0 \,
4 (3; 1, 0, 0, 0) \scriptstyle x^3 + 0 x^2 + 0 x + 0 \,=\, 0 \, \scriptstyle x^3 \,=\, 0 \,
5 (1; 1, -3) \scriptstyle x - 3 \,=\, 0 \, \scriptstyle x - 3 \,=\, 0 \, 3
5 (1; 1, +3) \scriptstyle x + 3 \,=\, 0 \, \scriptstyle x + 3 \,=\, 0 \, -3
5 (1; 2, -2) \scriptstyle 2x - 2 \,=\, 0 \, \scriptstyle 2 (x - 1) \,=\, 0 \,
5 (1; 2, +2) \scriptstyle 2x + 2 \,=\, 0 \, \scriptstyle 2 (x + 1) \,=\, 0 \,
5 (1; 3, -1) \scriptstyle 3x - 1 \,=\, 0 \, \scriptstyle 3x - 1 \,=\, 0 \, 1/3
5 (1; 3, +1) \scriptstyle 3x + 1 \,=\, 0 \, \scriptstyle 3x + 1 \,=\, 0 \, -1/3
5 (1; 4, 0) \scriptstyle 4x + 0 \,=\, 0 \, \scriptstyle 4x \,=\, 0 \,
5 (2; 1, 0, -2) \scriptstyle x^2 + 0 x - 2 \,=\, 0 \, \scriptstyle x^2 - 2 \,=\, 0 \, \scriptstyle \sqrt{2},\, -\sqrt{2} \,
5 (2; 1, 0, +2) \scriptstyle x^2 + 0 x + 2 \,=\, 0 \, \scriptstyle x^2 + 2 \,=\, 0 \, \scriptstyle \sqrt{2}~i,\, -\sqrt{2}~i \,
5 (2; 1, -1, -1) \scriptstyle x^2 - x - 1 \,=\, 0 \, \scriptstyle x^2 - x - 1 \,=\, 0 \, \scriptstyle \frac{1}{2} + \frac{\sqrt{5}}{2},\, \frac{1}{2} - \frac{\sqrt{5}}{2} \,
5 (2; 1, -1, +1) \scriptstyle x^2 - x + 1 \,=\, 0 \, \scriptstyle x^2 - x + 1 \,=\, 0 \, \scriptstyle \big( \frac{1}{2} + \frac{\sqrt{3}}{2}~i,\, \frac{1}{2} - \frac{\sqrt{3}}{2}~i \big) \,=\, \big( e^{\frac{i  \pi}{3}},\, e^{\frac{i 5 \pi}{3}} \big)\,
5 (2; 1, +1, -1) \scriptstyle x^2 + x - 1 \,=\, 0 \, \scriptstyle x^2 + x - 1 \,=\, 0 \, \scriptstyle -\frac{1}{2} + \frac{\sqrt{5}}{2},\, -\frac{1}{2} - \frac{\sqrt{5}}{2} \,
5 (2; 1, +1, +1) \scriptstyle x^2 + x + 1 \,=\, 0 \, \scriptstyle x^2 + x + 1 \,=\, 0 \, \scriptstyle \big( -\frac{1}{2} + \frac{\sqrt{3}}{2}~i,\, -\frac{1}{2} - \frac{\sqrt{3}}{2}~i \big) \,=\, \big( e^{\frac{i 2 \pi}{3}},\, e^{\frac{i 4 \pi}{3}} \big) \,
5 (2; 1, -2, 0) \scriptstyle x^2 - 2x + 0 \,=\, 0 \, \scriptstyle x (x - 2) \,=\, 0 \,
5 (2; 1, +2, 0) \scriptstyle x^2 + 2x + 0 \,=\, 0 \, \scriptstyle x (x + 2) \,=\, 0 \,
5 (2; 2, 0, -1) \scriptstyle 2x^2 + 0x - 1 \,=\, 0 \, \scriptstyle 2x^2 - 1 \,=\, 0 \, \scriptstyle \frac{\sqrt{2}}{2},\, -\frac{\sqrt{2}}{2} \,
5 (2; 2, 0, +1) \scriptstyle 2x^2 + 0x + 1 \,=\, 0 \, \scriptstyle 2x^2 + 1 \,=\, 0 \, \scriptstyle \frac{\sqrt{2}}{2}~i,\, -\frac{\sqrt{2}}{2}~i \,
5 (2; 2, -1, 0) \scriptstyle 2x^2 - x + 0 \,=\, 0 \, \scriptstyle x (2x - 1) \,=\, 0 \,
5 (2; 2, +1, 0) \scriptstyle 2x^2 + x + 0 \,=\, 0 \, \scriptstyle x (2x + 1) \,=\, 0 \,
5 (2; 3, 0, 0) \scriptstyle 3x^2 + 0x + 0 \,=\, 0 \, \scriptstyle 3x^2 \,=\, 0 \,
5 (3; 1, 0, 0, -1) \scriptstyle x^3 + 0x^2 + 0x - 1 \,=\, 0 \, \scriptstyle (x - 1) (x^2 + x + 1) \,=\, 0 \,
5 (3; 1, 0, 0, +1) \scriptstyle x^3 + 0x^2 + 0x + 1 \,=\, 0 \, \scriptstyle (x + 1) (x^2 - x + 1) \,=\, 0 \,
5 (3; 1, 0, -1, 0) \scriptstyle x^3 + 0x^2 - x + 0 \,=\, 0 \, \scriptstyle x (x - 1) (x + 1) \,=\, 0 \,
5 (3; 1, 0, +1, 0) \scriptstyle x^3 + 0x^2 + x + 0 \,=\, 0 \, \scriptstyle x (x^2 + 1) \,=\, 0 \,
5 (3; 1, -1, 0, 0) \scriptstyle x^3 - x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (x - 1) \,=\, 0 \,
5 (3; 1, +1, 0, 0) \scriptstyle x^3 + x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (x + 1) \,=\, 0 \,
5 (3; 2, 0, 0, 0) \scriptstyle 2x^3 + 0x^2 + 0x + 0 \,=\, 0 \, \scriptstyle 2x^3 \,=\, 0 \,
6 (1; 1, -4) \scriptstyle x - 4 \,=\, 0 \, \scriptstyle x - 4 \,=\, 0 \, 4
6 (1; 1, +4) \scriptstyle x + 4 \,=\, 0 \, \scriptstyle x + 4 \,=\, 0 \, -4
6 (1; 2, -3) \scriptstyle 2x - 3 \,=\, 0 \, \scriptstyle 2x - 3 \,=\, 0 \, 3/2
6 (1; 2, +3) \scriptstyle 2x + 3 \,=\, 0 \, \scriptstyle 2x + 3 \,=\, 0 \, -3/2
6 (1; 3, -2) \scriptstyle 3x - 2 \,=\, 0 \, \scriptstyle 3x - 2 \,=\, 0 \, 2/3
6 (1; 3, +2) \scriptstyle 3x + 2 \,=\, 0 \, \scriptstyle 3x + 2 \,=\, 0 \, -2/3
6 (1; 4, -1) \scriptstyle 4x - 1 \,=\, 0 \, \scriptstyle 4x - 1 \,=\, 0 \, 1/4
6 (1; 4, +1) \scriptstyle 4x + 1 \,=\, 0 \, \scriptstyle 4x + 1 \,=\, 0 \, -1/4
6 (1; 5, 0) \scriptstyle 5x + 0 \,=\, 0 \, \scriptstyle 5x \,=\, 0 \,
6 (2; 1, 0, -3) \scriptstyle x^2 + 0x - 3 \,=\, 0 \, \scriptstyle x^2 - 3 \,=\, 0 \, \scriptstyle \sqrt{3},\, -\sqrt{3} \,
6 (2; 1, 0, +3) \scriptstyle x^2 + 0x + 3 \,=\, 0 \, \scriptstyle x^2 + 3 \,=\, 0 \, \scriptstyle \sqrt{3}~i,\, -\sqrt{3}~i \,
6 (2; 1, -1, -2) \scriptstyle x^2 - x - 2 \,=\, 0 \, \scriptstyle (x - 2) (x + 1) \,=\, 0 \,
6 (2; 1, -1, +2) \scriptstyle x^2 - x + 2 \,=\, 0 \, \scriptstyle x^2 - x + 2 \,=\, 0 \, \scriptstyle \frac{1}{2} + \frac{\sqrt{7}}{2}~i,\, \frac{1}{2} - \frac{\sqrt{7}}{2}~i \,
6 (2; 1, +1, -2) \scriptstyle x^2 + x - 2 \,=\, 0 \, \scriptstyle (x + 2) (x - 1) \,=\, 0 \,
6 (2; 1, +1, +2) \scriptstyle x^2 + x + 2 \,=\, 0 \, \scriptstyle x^2 + x + 2 \,=\, 0 \, \scriptstyle -\frac{1}{2} + \frac{\sqrt{7}}{2}~i,\, -\frac{1}{2} - \frac{\sqrt{7}}{2}~i \,
6 (2; 1, -2, -1) \scriptstyle x^2 - 2x - 1 \,=\, 0 \, \scriptstyle x^2 - 2x - 1 \,=\, 0 \, \scriptstyle 1 + \sqrt{2},\, 1 - \sqrt{2} \,
6 (2; 1, -2, +1) \scriptstyle x^2 - 2x + 1 \,=\, 0 \, \scriptstyle (x - 1)^2 \,=\, 0 \,
6 (2; 1, +2, -1) \scriptstyle x^2 + 2x - 1 \,=\, 0 \, \scriptstyle x^2 + 2x - 1 \,=\, 0 \, \scriptstyle -1 + \sqrt{2},\, -1 - \sqrt{2} \,
6 (2; 1, +2, +1) \scriptstyle x^2 + 2x + 1 \,=\, 0 \, \scriptstyle (x + 1)^2 \,=\, 0 \,
6 (2; 1, -3, 0) \scriptstyle x^2 - 3x + 0 \,=\, 0 \, \scriptstyle x (x - 3) \,=\, 0 \,
6 (2; 1, +3, 0) \scriptstyle x^2 + 3x + 0 \,=\, 0 \, \scriptstyle x (x + 3) \,=\, 0 \,
6 (2; 2, -1, -1) \scriptstyle 2x^2 - x - 1 \,=\, 0 \, \scriptstyle (x - 1) (2x + 1) \,=\, 0 \,
6 (2; 2, -1, +1) \scriptstyle 2x^2 - x + 1 \,=\, 0 \, \scriptstyle 2x^2 - x + 1 \,=\, 0 \, \scriptstyle \frac{1}{4} + \frac{\sqrt{7}}{4}~i,\, \frac{1}{4} - \frac{\sqrt{7}}{4}~i \,
6 (2; 2, +1, -1) \scriptstyle 2x^2 + x - 1 \,=\, 0 \, \scriptstyle (2x - 1) (x + 1) \,=\, 0 \,
6 (2; 2, +1, +1) \scriptstyle 2x^2 + x + 1 \,=\, 0 \, \scriptstyle 2x^2 + x + 1 \,=\, 0 \, \scriptstyle -\frac{1}{4} + \frac{\sqrt{7}}{4}~i,\, -\frac{1}{4} - \frac{\sqrt{7}}{4}~i \,
6 (2; 2, -2, 0) \scriptstyle 2x^2 - 2x + 0 \,=\, 0 \, \scriptstyle 2x (x - 1) \,=\, 0 \,
6 (2; 2, +2, 0) \scriptstyle 2x^2 + 2x + 0 \,=\, 0 \, \scriptstyle 2x (x + 1) \,=\, 0 \,
6 (2; 3, 0, -1) \scriptstyle 3x^2 + 0x - 1 \,=\, 0 \, \scriptstyle 3x^2 - 1 \,=\, 0 \, \scriptstyle \frac{\sqrt{3}}{3},\, -\frac{\sqrt{3}}{3} \,
6 (2; 3, 0, +1) \scriptstyle 3x^2 + 0x + 1 \,=\, 0 \, \scriptstyle 3x^2 + 1 \,=\, 0 \, \scriptstyle \frac{\sqrt{3}}{3}~i,\, -\frac{\sqrt{3}}{3}~i \,
6 (2; 3, -1, 0) \scriptstyle 3x^2 - x + 0 \,=\, 0 \, \scriptstyle x (3x - 1) \,=\, 0 \,
6 (2; 3, +1, 0) \scriptstyle 3x^2 + x + 0 \,=\, 0 \, \scriptstyle x (3x + 1) \,=\, 0 \,
6 (2; 4, 0, 0) \scriptstyle 4x^2 + 0x + 0 \,=\, 0 \, \scriptstyle 4x^2 \,=\, 0 \,
6 (3; 1, 0, 0, -2) \scriptstyle x^3 + 0x^2 + 0x - 2 \,=\, 0 \, \scriptstyle x^3 - 2 \,=\, 0 \, \scriptstyle \sqrt[3]{2},\, \sqrt[3]{2}~e^{i \frac{2 \pi}{3}},\, \sqrt[3]{2}~e^{i \frac{4 \pi}{3}} \,
6 (3; 1, 0, 0, +2) \scriptstyle x^3 + 0x^2 + 0x + 2 \,=\, 0 \, \scriptstyle x^3 + 2 \,=\, 0 \, \scriptstyle \sqrt[3]{2}~e^{i \frac{\pi}{3}},\, -\sqrt[3]{2},\, \sqrt[3]{2}~e^{i \frac{5 \pi}{3}} \,
6 (3; 1, 0, -1, -1) \scriptstyle x^3 + 0x^2 - x - 1 \,=\, 0 \, \scriptstyle (x - 1) (x^2 + x + 1) \,=\, 0 \,
6 (3; 1, 0, -1, +1) \scriptstyle x^3 + 0x^2 - x + 1 \,=\, 0 \, \scriptstyle (x + 1) (x^2 - x + 1) \,=\, 0 \,
6 (3; 1, 0, +1, -1) \scriptstyle x^3 + 0x^2 + x - 1 \,=\, 0 \, \scriptstyle x^3 + x - 1 \,=\, 0 \,  ???[1]
6 (3; 1, 0, +1, +1) \scriptstyle x^3 + 0x^2 + x + 1 \,=\, 0 \, \scriptstyle x^3 + x + 1 \,=\, 0 \,  ???[1]
6 (3; 1, 0, -2, 0) \scriptstyle x^3 + 0x^2 - 2x + 0 \,=\, 0 \, \scriptstyle x (x^2 - 2) \,=\, 0 \,
6 (3; 1, 0, +2, 0) \scriptstyle x^3 + 0x^2 + 2x + 0 \,=\, 0 \, \scriptstyle x (x^2 + 2) \,=\, 0 \,
6 (3; 1, -1, 0, -1) \scriptstyle x^3 - x^2 + 0x - 1 \,=\, 0 \, \scriptstyle x^3 - x^2 - 1 \,=\, 0 \,  ???[1]
6 (3; 1, -1, 0, +1) \scriptstyle x^3 - x^2 + 0x + 1 \,=\, 0 \, \scriptstyle x^3 - x^2 + 1 \,=\, 0 \,  ???[1]
6 (3; 1, +1, 0, -1) \scriptstyle x^3 + x^2 + 0x - 1 \,=\, 0 \, \scriptstyle x^3 + x^2 - 1 \,=\, 0 \,  ???[1]
6 (3; 1, +1, 0, +1) \scriptstyle x^3 + x^2 + 0x + 1 \,=\, 0 \, \scriptstyle x^3 + x^2 + 1 \,=\, 0 \,  ???[1]
6 (3; 1, -1, -1, 0) \scriptstyle x^3 - x^2 - x + 0 \,=\, 0 \, \scriptstyle x (x^2 - x - 1) \,=\, 0 \,
6 (3; 1, -1, +1, 0) \scriptstyle x^3 - x^2 + x + 0 \,=\, 0 \, \scriptstyle x (x^2 - x + 1) \,=\, 0 \,
6 (3; 1, +1, -1, 0) \scriptstyle x^3 + x^2 - x + 0 \,=\, 0 \, \scriptstyle x (x^2 + x - 1) \,=\, 0 \,
6 (3; 1, +1, +1, 0) \scriptstyle x^3 + x^2 + x + 0 \,=\, 0 \, \scriptstyle x (x^2 + x + 1) \,=\, 0 \,
6 (3; 1, -2, 0, 0) \scriptstyle x^3 - 2x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (x - 2) \,=\, 0 \,
6 (3; 1, +2, 0, 0) \scriptstyle x^3 + 2x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (x + 2) \,=\, 0 \,
6 (3; 2, 0, 0, -1) \scriptstyle 2x^3 + 0x^2 + 0x - 1 \,=\, 0 \, \scriptstyle 2x^3 - 1 \,=\, 0 \, \scriptstyle \frac{1}{\sqrt[3]{2}},\, \frac{1}{\sqrt[3]{2}}~e^{i \frac{2 \pi}{3}},\, \frac{1}{\sqrt[3]{2}}~e^{i \frac{4 \pi}{3}} \,
6 (3; 2, 0, 0, +1) \scriptstyle 2x^3 + 0x^2 + 0x + 1 \,=\, 0 \, \scriptstyle 2x^3 + 1 \,=\, 0 \, \scriptstyle \frac{1}{\sqrt[3]{2}}~e^{i \frac{\pi}{3}},\, -\frac{1}{\sqrt[3]{2}},\, \frac{1}{\sqrt[3]{2}}~e^{i \frac{5 \pi}{3}} \,
6 (3; 2, 0, -1, 0) \scriptstyle 2x^3 + 0x^2 - x + 0 \,=\, 0 \, \scriptstyle x (2x - 1) \,=\, 0 \,
6 (3; 2, 0, +1, 0) \scriptstyle 2x^3 + 0x^2 + x + 0 \,=\, 0 \, \scriptstyle x (2x + 1) \,=\, 0 \,
6 (3; 2, -1, 0, 0) \scriptstyle 2x^3 - x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (2x - 1) \,=\, 0 \,
6 (3; 2, +1, 0, 0) \scriptstyle 2x^3 + x^2 + 0x + 0 \,=\, 0 \, \scriptstyle x^2 (2x + 1) \,=\, 0 \,


The algebraic numbers can thus be enumerated as (Cf. A??????; note that algebraic number sequences are not in the OEIS!)

\scriptstyle \{0,\, 1,\, -1,\, 2,\, -2,\, \frac{1}{2},\, -\frac{1}{2},\, i,\, -i,\, 3,\, -3,\, \frac{1}{3},\, -\frac{1}{3},\, \sqrt{2},\, -\sqrt{2},\, \sqrt{2}~i,\, -\sqrt{2}~i,\, \frac{1}{2} + \frac{\sqrt{5}}{2},\, \frac{1}{2} - \frac{\sqrt{5}}{2},\, \frac{1}{2} + \frac{\sqrt{3}}{2}~i,\, \frac{1}{2} - \frac{\sqrt{3}}{2}~i,\, -\frac{1}{2} + \frac{\sqrt{5}}{2},\, -\frac{1}{2} - \frac{\sqrt{5}}{2},\, -\frac{1}{2} + \frac{\sqrt{3}}{2}~i,\, -\frac{1}{2} - \frac{\sqrt{3}}{2}~i,\, \frac{\sqrt{2}}{2},\, -\frac{\sqrt{2}}{2},\, \frac{\sqrt{2}}{2}~i,\, -\frac{\sqrt{2}}{2}~i,\, 4,\, -4,\, \frac{3}{2},\, -\frac{3}{2},\, \frac{2}{3},\, -\frac{2}{3},\, \frac{1}{4},\, -\frac{1}{4},\, \ldots \} \,

Graded orderings of rational numbers

The algebraic numbers of degree \scriptstyle 1 \, are the roots of the polynomial equation

\sum_{i=0}^{1} a_{i} x^{i} = a_{1} x^{1} + a_{0} x^{0} = 0, \,

with \scriptstyle a_{1} \,\ge\, 1. \,

This gives graded orderings of rational numbers.

See also

Notes

  1. 1.0 1.1 1.2 1.3 1.4 1.5 Complicated cubic roots obtainable by the cubic formula.
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