Algebraic numbers

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The algebraic numbers are the roots of a nonconstant polynomial equation with integer coefficients (i.e. $\scriptstyle \{x\,|\,P(x)\,=\,0,\,P(x)\,\in \,\mathbb {Z} [x],\,P(x)\,=\,\omega (1)\}$)

$P(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots +a_{2}x^{2}+a_{1}x^{1}+a_{0}x^{0}=0,$

with $\scriptstyle n\,\geq \,1,\,a_{i}\,\in \,\mathbb {Z} ,\,a_{n}\,\geq \,1.$ The polynomial is said to be primitive when $\scriptstyle \gcd(a_{n},\,a_{n-1},\,\ldots ,\,a_{1},\,a_{0})\,=\,1$. Equivalently, the algebraic numbers are the roots of a nonconstant monic polynomial equation with rational coefficients in $\scriptstyle \mathbb {Q} [x]$.

For example, the numbers 3, $\scriptstyle {\frac {-1-{\sqrt {-23}}}{4}}\,$ and $\scriptstyle {\frac {-1+{\sqrt {-23}}}{4}}\,$ are all algebraic numbers, as they are the roots of the equation $4x^{3}-10x^{2}-18=0$.

The minimal polynomial for an algebraic number $\scriptstyle \alpha$ is the primitive polynomial of minimal degree which has $\scriptstyle \alpha$ as a root (the polynomial is thus irreducible, i.e. cannot be factored into polynomials, with integer coefficients, of lower degree). For example, $\scriptstyle {\frac {\sqrt {-5}}{2}}$ is an algebraic number, and $4x^{2}+5=0$ is its minimal polynomial.

If the polynomial with integer coefficients is monic, i.e. the leading coefficient $\scriptstyle a_{n}\,=\,1$, then the polynomial roots are algebraic integers. (The algebraic integers of degree 1 are the "linear integers," referred to as the rational integers in number theory (the ring of integers within the field of rational numbers), and commonly called the integers $\scriptstyle \mathbb {Z} \,$.)

Algebraic numbers by degree of minimal polynomial

1. Rational numbers: algebraic numbers of degree one]] (rational integers: 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)

Graded orderings of algebraic numbers

See: Orderings of algebraic numbers.

Arithmetic numbers

Arithmetic numbers are algebraic numbers which can be expressed with a finite number of algebraic operations, which consist of field operations (+, −, ×, /) and exponentiation with a [constant] rational exponent (i.e. powers and/or root extractions).

An algebraic number which can be expressed as a finite sequence of addition, subtraction, multiplication, division, exponentiation (with integer exponent) and roots (with integer index, e.g. square roots, cubic roots, etc.) in terms of natural numbers is sometimes called an "arithmetic number."

For example

${\frac {{\sqrt[{101}]{19}}+127{\sqrt[{29}]{257+{\sqrt[{67}]{13+{\sqrt {-71}}}}}}}{87-64{\sqrt[{997}]{97}}}},$

is an arithmetic number (an "explicit" closed-form algebraic number).

The arithmetic numbers are a proper subset of the algebraic numbers. Although all algebraic numbers of degree up to 4 are arithmetic, not all algebraic numbers of degree 5 and above are arithmetic. For example, the roots of $x^{5}+x+1=0$ are not arithmetic. (The roots of $p(x)=x^{5}+ax^{3}+bx^{2}+cx+d=0$, if $p(x)$ is irreducible [i.e. cannot factored into polynomials, with integer coefficients, of lower degree], and if $a$ and $b$ are even and $c$ and $d$ are odd, are not arithmetic.)

In general the roots of a polynomial are arithmetic if and only if the Galois group of the extension field of the polynomial over $\scriptstyle \mathbb {Q} \,$ is solvable.

Transcendental numbers

Numbers which are not the root of any polynomial with integer coefficients are called transcendental numbers. The algebraic numbers are a countable (denumerable) subset of the complex numbers, thus almost all numbers are transcendental. Examples include $\scriptstyle 47\pi \,$.

See also

• Orderings
• Transcendental numbers

External links

• Leo Goldmakher (U of T), UNDER CONSTRUCTION! (...) visualization of the field of algebraic numbers in the complex plane (...) (Graphic by Stephen J. Brooks.)
• Robert Munafo, RIES - Find Algebraic Equations, Given Their Solution.