All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a natural number, satisfies the above definition because is the root of a non-zero polynomial, namely.
The quadratic surds are algebraic numbers. If the quadratic polynomial is monic then the roots are further qualified as quadratic integers.
Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of th roots gives another algebraic number.
Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of th roots. This happens with many, but not all, polynomials of degree 5 or higher.
Gaussian integers: those complex numbers where both and are integers and are also quadratic integers.
* The numbers and are algebraic since they are roots of polynomials and, respectively.
* The golden ratio is algebraic since it is a root of the polynomial.
* The numbers Pi| and e | are not algebraic numbers.
Properties
Given an algebraic number, there is a uniquemonic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree, then the algebraic number is said to be of degree . For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
The set of algebraic numbers is countable, and therefore its Lebesgue measure as a subset of the complex numbers is 0. That is to say, "almost all" real and complex numbers are transcendental.
The sum, difference, product and quotient of two algebraic numbers is again algebraic, and the algebraic numbers therefore form a field . 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. The set of real algebraic numbers itself forms a field.
Related fields
Numbers defined by radicals
All numbers that can be obtained from the integers using a finite number of complex additions, subtractions, multiplications, divisions, and taking th roots where is a positive integer, are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory. An example is, where the unique real root is where is the generalized hypergeometric function.
Closed-form number
Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as or ln 2.
Algebraic integers
An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1. Examples of algebraic integers are, and. Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers. 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 that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If is a number field, its ring of integers is the subring of algebraic integers in, and is frequently denoted as. These are the prototypical examples of Dedekind domains.
