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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217101.png" />,
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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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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217102.png" /></td> </tr></table>
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0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}}
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217103.png" /> is the unit element of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217104.png" />, then the smallest such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217105.png" /> is a prime number; it is called the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217106.png" />. If there are no such numbers, then one says that the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217107.png" /> is zero or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217108.png" /> is a field of characteristic zero. Sometimes such a field is said to be without characteristic or of characteristic infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c0217109.png" />. Every field of characteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c02171010.png" /> contains a subfield isomorphic to the field of residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021710/c02171011.png" />.
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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, 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
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article