The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, ${\displaystyle (1+{\sqrt {5}})/2}$, is an algebraic number, because it is a root of the polynomial x² − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number ${\displaystyle 1+i}$ is algebraic because it is a root of x⁴ + 4.

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.

Examples

• All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer a and a (non-zero) natural number b, satisfies the above definition, because x = a/b is the root of a non-zero polynomial, namely bx − a.^[1]
• Quadratic irrational numbers, irrational solutions of a quadratic polynomial ax² + bx + c with integer coefficients a, b, and c, are algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified as quadratic integers.
  • Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers. This is because a + bi and a − bi are the two roots of the quadratic x² − 2ax + a² + b².
• A constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (By designating cardinal directions for +1, −1, +i, and −i, complex numbers such as ${\displaystyle 3+i{\sqrt {2}}}$ are considered constructible.)
• Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of nth roots gives another algebraic number.
• Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots (such as the roots of x⁵ − x + 1). That happens with many but not all polynomials of degree 5 or higher.
• Values of trigonometric functions of rational multiples of π (except when undefined): for example, cos π/7, cos 3π/7, and cos 5π/7 satisfy 8x³ − 4x² − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, and tan 15π/16 satisfy the irreducible polynomial x⁴ − 4x³ − 6x² + 4x + 1 = 0, and so are conjugate algebraic integers. This is the equivalent of angles which, when measured in degrees, have rational numbers.
• Some but not all irrational numbers are algebraic:
  • The numbers ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle {\frac {\sqrt[{3}]{3}}{2}}}$ are algebraic since they are roots of polynomials x² − 2 and 8x³ − 3, respectively.
  • The golden ratio φ is algebraic since it is a root of the polynomial x² − x − 1.
  • The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem).^[2]

Properties

Algebraic numbers on the complex plane colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4)

• If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
• Given an algebraic number, there is 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. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
• The algebraic numbers are dense in the reals. This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
• The set of algebraic numbers is countable (enumerable),^[3][4] and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, "almost all" real and complex numbers are transcendental.
• All algebraic numbers are computable and therefore definable and arithmetical.
• For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic.^[5]

Field

Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.

The sum, difference, product and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic, as can be demonstrated by using the resultant, and algebraic numbers thus form a field ${\displaystyle {\overline {\mathbb {Q} }}}$ (sometimes denoted by ${\displaystyle \mathbb {A} }$, but that usually denotes the adele ring). Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. That 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 so it is called the algebraic closure of the rationals.

The set of real algebraic numbers itself forms a field.^[6]

Related fields

Algebraic integers

Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer)

An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are ${\displaystyle 5+13{\sqrt {2}},}$ ${\displaystyle 2-6i,}$ and ${\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}$ Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials x − k for all ${\displaystyle k\in \mathbb {Z} }$. 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 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

• Algebraic solution
• Gaussian integer
• Eisenstein integer
• Quadratic irrational number
• Fundamental unit
• Root of unity
• Gaussian period
• Pisot–Vijayaraghavan number
• Salem number

Notes

References

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 0-13-004763-5, MR 1129886
• Hardy, Godfrey Harold; Wright, Edward M. (1972), An introduction to the theory of numbers (5th ed.), Oxford: Clarendon, ISBN 0-19-853171-0
• Ireland, Kenneth; Rosen, Michael (1990) [1st ed. 1982], A Classical Introduction to Modern Number Theory (2nd ed.), Berlin: Springer, doi:10.1007/978-1-4757-2103-4, ISBN 0-387-97329-X, MR 1070716
• Lang, Serge (2002) [1st ed. 1965], Algebra (3rd ed.), New York: Springer, ISBN 978-0-387-95385-4, MR 1878556
• Niven, Ivan M. (1956), Irrational Numbers, Mathematical Association of America
• Ore, Øystein (1948), Number Theory and Its History, New York: McGraw-Hill