# Difference between revisions of "Characteristic of a field"

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− | A positive prime number or the number 0 that is uniquely determined for a given field in the following way. If for some | + | A positive prime number or the number 0 that is uniquely determined for a given field in the following way. If for some $n$, |

− | + | $$ | |

− | + | 0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}} | |

− | + | $$ | |

− | where | + | where $e$ is the unit element of the field $F$, then the smallest such $n$ is a prime number; it is called the characteristic of $F$. If there are no such numbers, then one says that the characteristic of $F$ is zero or that $F$ is a field of characteristic zero. Sometimes such a field is said to be without characteristic or of characteristic infinity $(\infty)$. Every field of characteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to the field of residue classes modulo $p$. |

## Revision as of 19:35, 30 August 2013

A positive prime number or the number 0 that is uniquely determined for a given field in the following way. If for some $n$, $$ 0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}} $$ where $e$ is the unit element of the field $F$, then the smallest such $n$ is a prime number; it is called the characteristic of $F$. If there are no such numbers, then one says that the characteristic of $F$ is zero or that $F$ is a field of characteristic zero. Sometimes such a field is said to be without characteristic or of characteristic infinity $(\infty)$. Every field of characteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to the field of residue classes modulo $p$.

**How to Cite This Entry:**

Characteristic of a field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Characteristic_of_a_field&oldid=17329