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 ,

where is the unit element of the field , then the smallest such is a prime number; it is called the characteristic of . If there are no such numbers, then one says that the characteristic of is zero or that is a field of characteristic zero. Sometimes such a field is said to be without characteristic or of characteristic infinity . Every field of characteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic contains a subfield isomorphic to the field of residue classes modulo .

How to Cite This Entry:
Characteristic of a field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article