nLab algebraic number




An algebraic number is a root of a polynomial with integer coefficients (or, equivalently, with rational coeffients). Equivalently, an element α\alpha of a field extension KK of the rational numbers \mathbb{Q} is algebraic if the subfield (α)\mathbb{Q}(\alpha) is a finite degree extension, i.e., is finite-dimensional as a vector space over \mathbb{Q}.

Since the rational numbers are a subfield of the complex numbers, and since the complex numbers are an algebraically closed field, algebraic numbers are naturally regarded as a sub-field of complex numbers

. \mathbb{Q} \hookrightarrow \mathbb{C} \,.


Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic…). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, +i+i near top.

due to Stephen J. Brooks here

But the collection of all algebraic numbers forms itself already an algebraically closed field, typically denoted ¯\overline{\mathbb{Q}}, as this is the algebraic closure of the field \mathbb{Q} of rational numbers. This also follows easily from the equivalent definition of algebraic numbers in terms of finite degree extensions. The absolute Galois group Gal(¯,)Gal(\overline{\mathbb{Q}}, \mathbb{Q}) is peculiar, see there.

A number α\alpha (especially a complex number) is called transcendental if the distance from that number to every algebraic number is positive:

isTranscendental(α)β¯.|αβ|>0\mathrm{isTranscendental}(\alpha) \coloneqq \forall \beta \in \overline{\mathbb{Q}}.\vert \alpha - \beta \vert \gt 0

Famous examples are the base (e=2.7\mathrm{e} = 2.7\ldots ) and period (2πi=6.28i2 \pi \mathrm{i} = 6.28\ldots \mathrm{i}, or equivalently π=3.14\pi = 3.14\ldots ) of the natural logarithm.

Given a field kk, an algebraic number field KK over kk is a finite-degree extension of kk. By default, the term “algebraic number field” means an algebraic number field over the rational numbers. If α\alpha is an algebraic number over \mathbb{Q} then [α]\mathbb{Q}[\alpha] is a number field, however the field of all algebraic numbers is not a number field.

An algebraic integer is a root of a monic polynomial with integer coefficients. Equivalently, an element α\alpha of a field extension KK of \mathbb{Q} is an algebraic integer if the ring [α]\mathbb{Z}[\alpha] is of finite rank as a \mathbb{Z}-module. It follows easily from this characterization that the collection of all algebraic integers forms a commutative ring.

In p-adic complex numbers

A p-adic complex number α p¯\alpha \in \overline{\mathbb{C}_p} is called transcendental if the p-adic norm of the difference of α\alpha and every algebraic number is positive:

isTranscendental(α)β¯.|αβ| p>0\mathrm{isTranscendental}(\alpha) \coloneqq \forall \beta \in \overline{\mathbb{Q}}.\vert \alpha - \beta \vert_p \gt 0


See also

Last revised on May 6, 2022 at 01:39:22. See the history of this page for a list of all contributions to it.