Number field

From Encyclopedia of Mathematics
Revision as of 15:55, 15 April 2014 by Ivan (talk | contribs) (TeX)
Jump to: navigation, search

A field consisting of complex (e.g., real) numbers. A set of complex numbers forms a number field if and only if it contains more than one element and with any two elements $\alpha$ and $\beta$ their difference $\alpha-\beta$ and quotient $\alpha/\beta$ ($\beta\neq0$). Every number field contains infinitely many elements. The field of rational numbers is contained in every number field.

Examples of number fields are the fields of rational numbers, real numbers, complex numbers, or Gaussian numbers (cf. Gauss number). The set of all numbers of the form $H(\alpha)/F(\alpha)$, $F(\alpha)\neq0$, forms a number field, $Q(\alpha)$, where $\alpha$ is a fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.


An algebraic number field $K$ of degree $n$ is an extension of degree $n$ of the field $\mathbf Q$ of rational numbers. Alternatively, a number field $K$ is an algebraic number field (of degree $n$) if every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. A number field that is not algebraic is called transcendental. (Cf. also Algebraic number theory; Extension of a field; Transcendental extension.)


[a1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9
How to Cite This Entry:
Number field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Shidlovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article