Algebraic number

Algebraic number over the field $\mathbb {F}$ ${\ displaystyle \ mathbb {F}}$ $\ mathbb {F}$ - an element of algebraic closure of the field $\mathbb {F}$ ${\ displaystyle \ mathbb {F}}$ $\ mathbb {F}$ , that is, the root of a polynomial (not identically equal to zero ) with coefficients from $\mathbb {F}$ ${\ displaystyle \ mathbb {F}}$ $\ mathbb {F}$ .

If the field is not specified, then the field of rational numbers is assumed, that is $\mathbb {F} =\mathbb {Q}$ ${\ displaystyle \ mathbb {F} = \ mathbb {Q}}$ ${\ displaystyle \ mathbb {F} = \ mathbb {Q}}$ , in this case the field of algebraic numbers is usually denoted $\mathbb {A}$ ${\ displaystyle \ mathbb {A}}$ $\ mathbb {A}$ . This article is devoted precisely to these "rational algebraic numbers." Field $\mathbb {A}$ ${\ displaystyle \ mathbb {A}}$ $\ mathbb {A}$ is a subfield of complex numbers .

Content

Related definitions

A real number or a complex number that is not algebraic is called transcendental .
Algebraic integers are the roots of polynomials with integer coefficients and with the highest coefficient equal to one.
If a $α$ {\ displaystyle \ alpha} Is an algebraic number, then among all polynomials with rational coefficients that have $α$ {\ displaystyle \ alpha} by its root, there exists a unique polynomial of the least degree with the highest coefficient equal to one. Such a polynomial is called a minimal or canonical algebraic number polynomial. $α$ {\ displaystyle \ alpha} (sometimes a canonical term is a polynomial obtained from a minimal multiplication by the least common multiple of its coefficients, that is, a polynomial with integer coefficients).
- The minimal polynomial is always irreducible .
- Degree of canonical polynomial $\alpha$ ${\ displaystyle \ alpha}$ called the degree of algebraic number $\alpha$ ${\ displaystyle \ alpha}$ .
- Other roots of the canonical polynomial $\alpha$ ${\ displaystyle \ alpha}$ are called conjugate with $\alpha$ ${\ displaystyle \ alpha}$ .
- Height of algebraic number $\alpha$ ${\ displaystyle \ alpha}$ is called the largest of the absolute values of the coefficients in an irreducible and primitive polynomial with integer coefficients, having $\alpha$ ${\ displaystyle \ alpha}$ its root.

Examples

Rational numbers , and only they, are algebraic numbers of the first degree.
Imaginary unit $i$ ${\ displaystyle i}$ and ${\sqrt {2}}$ ${\ displaystyle {\ sqrt {2}}}$ are algebraic numbers of the second degree. Related to them are respectively $-i$ ${\ displaystyle -i}$ and $-{\sqrt {2}}$ ${\ displaystyle - {\ sqrt {2}}}$ .
For any natural number $n$ ${\ displaystyle n}$ number ${\sqrt[{n}]{3}}$ ${\ displaystyle {\ sqrt [{n}] {3}}}$ is an algebraic number of degree $n$ ${\ displaystyle n}$ .

Properties

The set of algebraic numbers is countable , and therefore, its measure is zero.
The set of algebraic numbers is dense on the complex plane .
The sum, difference, product and quotient ^{[1] of} two algebraic numbers are algebraic numbers, that is, the set of all algebraic numbers forms a field .
The root of a polynomial with algebraic coefficients is an algebraic number, that is, the field of algebraic numbers is algebraically closed .
For every algebraic number $\alpha$ ${\ displaystyle \ alpha}$ there is such a natural $N$ ${\ displaystyle N}$ , what $N\alpha$ ${\ displaystyle N \ alpha}$ - an algebraic integer .
Algebraic number $\alpha$ ${\ displaystyle \ alpha}$ degrees $n$ ${\ displaystyle n}$ It has $n$ ${\ displaystyle n}$ various conjugate numbers (including yourself).
$\alpha$ ${\ displaystyle \ alpha}$ and $\beta$ ${\ displaystyle \ beta}$ are conjugate if and only if there is an automorphism of the field $\mathbb {A}$ ${\ displaystyle \ mathbb {A}}$ translating $\alpha$ ${\ displaystyle \ alpha}$ at $\beta$ ${\ displaystyle \ beta}$ .
Any algebraic number is computable and, therefore, arithmetic .
The order on the set of real algebraic numbers is isomorphic to the order on the set of rational numbers. ^{[ clear ]}

Numbers written with radicals

Any number that can be obtained from integers using the four actions of arithmetic (addition, subtraction, multiplication, division), as well as extracting the root of an integer degree, is algebraic. So, for example, the number will be algebraic. ${\sqrt {\frac {1998}{{\sqrt[{19}]{98}}-{\sqrt[{198}]{8}}}}}$ ${\ displaystyle {\ sqrt {\ frac {1998} {{\ sqrt [{19}] {98}} - {\ sqrt [{198}] {8}}}}}$ as well as numbers of the form $Q_{1}^{Q_{2}}+Q_{3}^{Q_{4}}+\ldots +Q_{n}^{Q_{n+1}}$ ${\ displaystyle Q_ {1} ^ {Q_ {2}} + Q_ {3} ^ {Q_ {4}} + \ ldots + Q_ {n} ^ {Q_ {n + 1}}}$ where $Q_{1},Q_{2},Q_{3},Q_{4}\dots Q_{n+1}$ ${\ displaystyle Q_ {1}, Q_ {2}, Q_ {3}, Q_ {4} \ dots Q_ {n + 1}}$ - rational numbers .

However, not all algebraic numbers can be written with the help of radicals. For example, according to the Abel – Ruffini theorem, there are polynomials of the fifth degree with integer coefficients that are not solvable in radicals. The roots of such a polynomial are algebraic numbers that cannot be constructed from four whole arithmetic operations and the extraction of roots ^[2] .

History

The name algebraic and transcendental numbers suggested by Euler in 1775. At that time, not a single transcendental number was known ^[2] . Algebraic fields other than rational, began to consider Gauss . In substantiating the theory of biquadratic residues, he developed the arithmetic of integer Gaussian numbers , that is, numbers of the form $a+bi$ ${\ displaystyle a + bi}$ where $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ - integers . The continuation of Gauss's research in the second half of the 19th century led to the construction of a general theory of algebraic numbers ^[3] . Further, studying the theory of cubic residues, Jacobi and Eisenstein created arithmetic of numbers of the form $a+b\rho$ ${\ displaystyle a + b \ rho}$ where $\rho =(-1+i{\sqrt {3}})/2$ ${\ displaystyle \ rho = (- 1 + i {\ sqrt {3}}) / 2}$ Is the cube root of one , and $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ - whole numbers. In 1844, Liouville proved a theorem on the impossibility of too good approximation of the roots of polynomials with rational coefficients by rational fractions, and, as a result, he introduced formal concepts of algebraic and transcendental (that is, all other real) numbers. Attempts to prove Fermat's great theorem led Kummer to study the fields of division of a circle , introduce the concept of an ideal, and create elements of the theory of algebraic numbers. In the works of Dirichlet , Kronecker , Hilbert and others, the theory of algebraic numbers was further developed. Russian mathematicians Zolotarev (the theory of ideals ), Voronoi (cubic irrationality, units of cubic fields), Markov (cubic field), Sokhotsky (theory of ideals) and others made a large contribution to it.

Notes

↑ except quotient by zero
↑ ¹ ² A. Zhukov. Algebraic and transcendental numbers // Kvant. - 1998. - № 4 .
↑ I. M. Vinogradov. Karl Friedrich Gauss // Works on Number Theory. - Moscow : USSR Academy of Sciences, 1959.

Links

Feldman, N. Algebraic and transcendental numbers // Kvant , № 7, 1983.
Nesterenko Yu. V. Lectures on algebraic numbers // Abstract of a course of lectures delivered at the Moscow State University .

[1] xcept quotient by zero

[kvant-2] ¹ ² A. Zhukov. Algebraic and transcendental numbers // Kvant. - 1998. - № 4 .

[3] I. M. Vinogradov. Karl Friedrich Gauss // Works on Number Theory. - Moscow : USSR Academy of Sciences, 1959.