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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 $n$,
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An invariant of a [[field]] which is either a [[prime number]] or the number zero, uniquely determined for a given field in the following way. If for some $n > 0$,
 
$$
 
$$
 
0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}}
 
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 $p$.
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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 $n$, 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 $p$: in each case this is the [[prime field]].

Revision as of 19:38, 30 August 2013

An invariant of a field which is either a prime number or the number zero, uniquely determined for a given field in the following way. If for some $n > 0$, $$ 0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}} $$ 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 $n$, 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 $p$: in each case this is the prime field.

How to Cite This Entry:
Characteristic of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_of_a_field&oldid=30274
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article