Algebraic number Totally Explained

Everything about Algebraic Number totally explained

In mathematics, an algebraic number is a complex number that's a root of a non-zero polynomial with rational (or equivalently, integer) coefficients. Complex numbers that are not algebraic are said to be transcendental.

Examples

The rational numbers, those expressed as the ratio of two whole numbers b and a, a not equal to zero, satisfy the above definition because x = −b/a is derived from (and satisfies) ax + b = 0. (In general, a or b can be negative, as can x).
Some irrational numbers are algebraic and some are not: ť * The numbers √2 and ³√3/2 are algebraic since they're the roots of x² − 2 = 0 and 8x³ − 3 = 0, respectively.

* The golden ratio φ is algebraic since it's a root of the polynomial x² − x − 1 = 0. ť * The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem) hence they're "transcendental".
The constructible numbers (those that, starting with a unit, can be constructed with ruler and compass, for example the square root of 2) are algebraic.
The quadratic surds (roots of a quadratic equation ax² + bx + c = 0 with integral coefficents a, b, and c) are algebraic numbers. Thus those complex numbers derived from ax² + bx + c = 0 — those corresponding to the case when the exponent n = 2 — are called quadratic numbers, or quadratic integers as the case may be.
Gaussian integers — those complex numbers a + bi where both a and b are integers are also quadratic integers.
When the lead coefficient for example a₀ is 1, the satisfactory x is/are said to be (an) algebraic integer(s). Note that an "algebraic integer" need not be a counting number such as 1, 2, 3, ... or a negative counterpart. ť * This definition comes from the notion that x = −b/a satisfies ax + b = 0, and when a = 1 then x = −b (for example b here being a positive or negative counting number or 0). But observe that from 1·x² + 4 = 0, x = 2i and −2i. So these two x are "algebraic integers" as well. This applies for any value of lead-exponent n. (See more below).

Properties

The set of algebraic numbers is countable (enumerable).

Hence, the set of algebraic numbers has Lebesgue measure zero (as a subset of the complex numbers), for example "almost all" complex numbers, are not algebraic. A complex number that isn't algebraic is called a transcendental number.

Given an algebraic number, there's a unique monic polynomial (with rational coefficients) of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number.

All algebraic numbers are computable and therefore definable.

The field of algebraic numbers

The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field, sometimes denoted by

mathbb

. It can be shown that every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. This can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it's the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
All the above statements are most easily proved in the general context of algebraic elements of a field extension.

Numbers defined by radicals

All numbers which can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking n^th roots (where n is a positive integer) are algebraic. The converse, however, isn't true: there are algebraic numbers which can't be obtained in this manner. All of these numbers are solutions to polynomials of degree ≥ 5. This is a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). An example of such a number is the unique real root of (which is approximately 1.167303978261418684256).

Algebraic integers

An algebraic integer is a number which is a root of a polynomial with integer coefficients (that is, an algebraic number) with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are 3√ + 5, 6i − 2 and (1 + i√)/2.
The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O_K. These are the prototypical examples of Dedekind domains.

Special classes of algebraic number

Pisot-Vijayaraghavan number

Salem number

Footnotes

Further Information

Get more info on 'Algebraic Number'.

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