# 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 and their difference and quotient (). 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 , , forms a number field, , where is a fixed complex number and and range over the polynomials with rational coefficients.

#### Comments

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

#### References

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

**How to Cite This Entry:**

Number field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Number_field&oldid=14270