Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently, by clearing denominators, with integer coefficients).

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.

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 complex numbers is uncountable, but the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the 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 $x = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} a / b$ is the root of a non-zero polynomial, namely $bx - a$ .[1]
Quadratic irrational numbers of a quadratic polynomial $ax 2 + bx + c$ 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.
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 $3+i{\sqrt {2}}$ are considered constructible.)
Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of $n$ th roots gives another algebraic number.
Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of $n$ th roots (such as the roots of $x 5 - x + 1$ ). That happens with many but not all polynomials of degree 5 or higher.
Gaussian integers, complex numbers $a + bi$ for which both $a$ and $b$ are integers and also quadratic integers.
Values of trigonometric functions of rational multiples of $π$ (except when undefined): that is, the trigonometric numbers such as $cos π / 7$ , $cos 3 π / 7$ , $cos 5 π / 7$ satisfies $8 x 3 - 4 x 2 - 4 x + 1 = 0$ . The polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, $tan 3 π / 16$ , $tan 7 π / 16$ , $tan 11 π / 16$ , $tan 15 π / 16$ satisfy the irreducible polynomial $x 4 - 4 x 3 - 6 x 2 + 4 x + 1 = 0$ , and so are conjugate algebraic integers.
Some but not all irrational number are algebraic:
- The numbers ${\sqrt {2}}$ and ${\frac {\sqrt[{3}]{3}}{2}}$ are algebraic since they are roots of polynomials $x 2 - 2$ and $8 x 3 - 3$ , respectively.
- The golden ratio $φ$ is algebraic since it is a root of the polynomial $x 2 - x - 1$ .
- 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 (red = 1, green = 2, blue = 3, yellow = 4)

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 real algebraic numbers are dense in the reals, linearly ordered, and without first or last element (and therefore order-isomorphic to the set of rational numbers).
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 $ℚ$ (sometimes denoted by $\mathbb {A}$ , 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

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 $n$ th roots where $n$ is a positive integer (radical expressions), 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 (see Quintic equations and the Abel–Ruffini theorem). An example is $x 5 - x - 1$ , where the unique real root is

{\begin{aligned}x&={\frac {1}{32}}\left(32\operatorname {_{4}F_{3}} \left(\left.{\begin{array}{ccc}-{\frac {1}{20}},{\frac {3}{20}},{\frac {7}{20}},{\frac {11}{20}}\\{\frac {1}{4}},{\frac {1}{2}},{\frac {3}{4}}\end{array}}\right|{\frac {3125}{256}}\right)+8\operatorname {_{4}F_{3}} \left(\left.{\begin{array}{ccc}{\frac {1}{5}},{\frac {2}{5}},{\frac {3}{5}},{\frac {4}{5}}\\{\frac {1}{2}},{\frac {3}{4}},{\frac {5}{4}}\end{array}}\right|{\frac {3125}{256}}\right)-5\operatorname {_{4}F_{3}} \left(\left.{\begin{array}{ccc}{\frac {9}{20}},{\frac {13}{20}},{\frac {17}{20}},{\frac {21}{20}}\\{\frac {3}{4}},{\frac {5}{4}},{\frac {3}{2}}\end{array}}\right|{\frac {3125}{256}}\right)+5\operatorname {_{4}F_{3}} \left(\left.{\begin{array}{ccc}{\frac {7}{10}},{\frac {9}{10}},{\frac {11}{10}},{\frac {13}{10}}\\{\frac {5}{4}},{\frac {3}{2}},{\frac {7}{4}}\end{array}}\right|{\frac {3125}{256}}\right)\right)\\[8pt]&=1.167303978261418684\ldots \end{aligned}}

where

\operatorname {_{p}F_{q}} \left(\left.{\begin{array}{ccc}a_{1},a_{2},\ldots ,a_{p}\\b_{1},b_{2},\ldots ,b_{q}\end{array}}\right|z\right)

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 $e$ or ln 2.

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 $5+13{\sqrt {2}},$ $2-6i,$ 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 $k \in ℤ.$ 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

Notes

Some of the following examples come from Hardy and Wright 1972: 159–160 and pp. 178–179
Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf. Hardy and Wright p. 161ff
Hardy and Wright 1972:160 / 2008:205
Niven 1956, Theorem 7.5.
Niven 1956, Corollary 7.3.
Niven (1956) p. 92.

References

Artin, Michael (1991), Algebra, Prentice Hall, ISBN 0-13-004763-5, MR 1129886
Hardy, G. H. and Wright, E. M. 1978, 2000 (with general index) An Introduction to the Theory of Numbers: 5th Edition, Clarendon Press, Oxford UK, ISBN 0-19-853171-0
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84 (Second ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2103-4, ISBN 0-387-97329-X, MR 1070716
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
Niven, Ivan 1956. Irrational Numbers, Carus Mathematical Monograph no. 11, Mathematical Association of America.
Ore, Øystein 1948, 1988, Number Theory and Its History, Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)

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[1] Some of the following examples come from Hardy and Wright 1972: 159–160 and pp. 178–179

[2] Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf. Hardy and Wright p. 161ff

[3] Hardy and Wright 1972:160 / 2008:205

[4] Niven 1956, Theorem 7.5.

[5] Niven 1956, Corollary 7.3.

[6] Niven (1956) p. 92.

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