# Number field

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.

#### Comments

The term "number field" is often used for the specific case of an algebraic number field: a finite degree extension of the field $\mathbf Q$ of rational numbers: 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.

#### References

[a1] | E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 Zbl 0115.03601 |

[b1] | H. Salzmann, T. Grundhöfer, H. Hähl, R. Löwen, "The classical fields. Structural features of the real and rational numbers" Encyclopedia of Mathematics and Its Applications 112 Cambridge University Press (2007) ISBN 0-521-86516-6 Zbl 1173.00006 |

**How to Cite This Entry:**

Number field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Number_field&oldid=54631