Algebraic number
An algebraic number is a number that is a
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
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 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.[2]
- Some but not all irrational numbers are algebraic:
- The numbers and 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).[3]
Properties
- 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 quadratic irrational.
- The algebraic numbers are 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),[4][5] 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 arithmetical.
- For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic.[6]
Field
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[7] (sometimes denoted by , 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.
Related fields
Numbers defined by radicals
Any number that can be obtained from the integers using a
has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations.
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 "
Algebraic integers
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 and Therefore, the algebraic integers constitute a proper
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 OK. 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
- ^ Some of the following examples come from Hardy & Wright (1972, pp. 159–160, 178–179)
- ^ Garibaldi 2008.
- ^ Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf. Hardy & Wright (1972, p. 161ff)
- ^ Hardy & Wright 1972, p. 160, 2008:205.
- ^ Niven 1956, Theorem 7.5..
- ^ Niven 1956, Corollary 7.3..
- ^ Niven 1956, p. 92.
References
- MR 1129886
- Garibaldi, Skip (June 2008), "Somewhat more than governors need to know about trigonometry", Mathematics Magazine, 81 (3): 191–200, JSTOR 27643106
- ISBN 0-19-853171-0
- Ireland, Kenneth; Rosen, Michael (1990) [1st ed. 1982], A Classical Introduction to Modern Number Theory (2nd ed.), Berlin: Springer, MR 1070716
- Lang, Serge (2002) [1st ed. 1965], Algebra (3rd ed.), New York: Springer, MR 1878556
- Niven, Ivan M. (1956), Irrational Numbers, Mathematical Association of America
- Ore, Øystein (1948), Number Theory and Its History, New York: McGraw-Hill