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Cubic numbers

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Not to be confused with cubes (also called perfect cubes), integers which are the third power of an integer.

Cubic numbers are the roots of the cubic equation

${\displaystyle a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0,}$

where ${\displaystyle \scriptstyle a_{3},\,a_{2},\,a_{1},\,a_{0}\,\in \,\mathbb {Z} ,\,a_{3}\,\geq \,1.}$

Although cubic numbers do include quadratic numbers and rational numbers as subsets, cubic numbers usually refer to numbers for which the minimal polynomial has degree 3.

Cubic integers

Cubic integers are the zeros of the monic cubic polynomial

${\displaystyle x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}$

where ${\displaystyle \scriptstyle a_{2},\,a_{1},\,a_{0}\,\in \,\mathbb {Z} .}$

Examples

Roots of x^3 - x^2 - 1

A092526 Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3. (Real root of ${\displaystyle \scriptstyle x^{3}-x^{2}-1}$.)

1.46557123187676...

Cubic numbers among the algebraic numbers

1. Rational numbers: algebraic numbers of degree one (rational integers:[1] algebraic integers of degree one)
2. Quadratic numbers: algebraic numbers of degree two (quadratic integers: algebraic integers of degree two)
3. Cubic numbers: algebraic numbers of degree three (cubic integers: algebraic integers of degree three)
4. Quartic numbers: algebraic numbers of degree four (quartic integers: algebraic integers of degree four)
5. Quintic numbers: algebraic numbers of degree five (quintic integers: algebraic integers of degree five)
6. ...

Notes

1. The rational integers are the common integers designated by ${\displaystyle \scriptstyle \mathbb {Z} }$.