# Difference between revisions of "Number field"

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− | A [[Field|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 | + | {{TEX|done}} |

+ | A [[Field|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|Gauss number]]). The set of all numbers of the form | + | Examples of number fields are the fields of rational numbers, real numbers, complex numbers, or Gaussian numbers (cf. [[Gauss number|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==== | ====Comments==== | ||

− | An algebraic number field | + | 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|Algebraic number theory]]; [[Extension of a field|Extension of a field]]; [[Transcendental extension|Transcendental extension]].) |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9</TD></TR></table> |

## Revision as of 15:55, 15 April 2014

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

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.)

#### 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=31745