Complex number that is a root of a non-zero polynomial in one variable with rational 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.
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, $(1+{\sqrt {5}})/2$, is an algebraic number, because it is a root of the polynomial x^{2} − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the
complex number$1+i$ is algebraic because it is a root of x^{4} + 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.
All
rational numbers are algebraic. Any rational number, expressed as the quotient of an
integera and a (non-zero)
natural numberb, 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 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.
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 x^{2} - 2ax + a^{2} + b^{2}.
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
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 x^{5} − 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 8x^{3} − 4x^{2} − 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 x^{4} − 4x^{3} − 6x^{2} + 4x + 1 = 0, and so are conjugate
algebraic integers.
Some but not all irrational numbers are algebraic:
The numbers ${\sqrt {2}}$ and ${\frac {\sqrt[{3}]{3}}{2}}$ are algebraic since they are roots of polynomials x^{2} − 2 and 8x^{3} − 3, respectively.
The
golden ratioφ is algebraic since it is a root of the polynomial x^{2} − x − 1.
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
densein 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.
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${\overline {\mathbb {Q} }}$ (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
Any number that can be obtained from the integers using a
finite number of
additions,
subtractions,
multiplications,
divisions, and taking (possibly complex) nth roots where n 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 (see
Quintic equations and the
Abel–Ruffini theorem). For example, the equation:
$x^{5}-x-1=0$
has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations.
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 $k\in \mathbb {Z}$. 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.
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, vol. 84 (Second ed.), Berlin, New York: Springer-Verlag,
doi:
10.1007/978-1-4757-2103-4,
ISBN0-387-97329-X,
MR1070716