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Encyclopedia > Algebraic number

In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. Without further qualification, it is assumed that an algebraic number is a complex number, but one can also consider algebraic numbers in other fields, such as fields of p-adic numbers. All these algebraic numbers belong to some algebraic number field. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... This article presents the essential definitions. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields...


All rationals are algebraic. An irrational number may or may not be algebraic. For example, 21/2 (the square root of 2) and 31/3/2 (half the cube root of 3) are algebraic because they are the solutions of x2 − 2 = 0 and 8x3 − 3 = 0, respectively. The imaginary unit i is algebraic, since it satisfies x2 + 1 = 0. In mathematics, an irrational number is any real number that is not a rational number, i. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... Plot of y = In mathematics, the cube root ( ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. ... In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ...


Numbers that are not algebraic are called transcendental numbers. Most complex numbers are transcendental, because the set of algebraic numbers is countable while the set of complex numbers, and therefore also the set of transcendental numbers, are not. Examples of transcendental numbers include π and e. Other examples are provided by the Gelfond-Schneider theorem. In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, a countable set is a set with the same cardinality (i. ... Lower-case Ï€ (the lower case letter is usually used for the constant) The mathematical constant Ï€ is an irrational number, approximately equal to 3. ... e is the unique number such that the derivative (slope) of f(x)=ex at any point is equal to the height of the function at that point. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ...


All algebraic numbers are computable and therefore definable. In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ...


If an algebraic number satisfies a polynomial equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number of degree n. In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...


The concept of algebraic numbers can be generalized to arbitrary field extensions; elements in such extensions that satify polynomial equations are called algebraic elements. In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ...

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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{A} or overline{mathbb{Q}}. 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 is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals. This article presents the essential definitions. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...


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 nth roots (where n is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers which cannot 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 would be the unique real root of x5 − x − 1 = 0. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... The Abel–Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...


Algebraic integers

Main article: algebraic integer

An algebraic number which satisfies a polynomial equation of degree n with leading coefficient an = 1 (that is, a monic polynomial) and all other coefficients ai belonging to the set Z of integers, is called an algebraic integer. Examples of algebraic integers are 3√2 + 5 and 6i - 2. In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...


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 OK. These are the prototypical examples of Dedekind domains. In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ...


Special classes of algebraic number


  Results from FactBites:
 
PlanetMath: algebraic number theory (953 words)
Algebraic number theory is the study of algebraic numbers, their properties and their applications.
The main object of study in algebraic number theory is the number field.
This is version 30 of algebraic number theory, born on 2005-03-15, modified 2006-03-07.
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