characteristic
Let be a field. The characteristic of is commonly given by one of three equivalent definitions:
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if there is some positive integer for which the result of adding any element to itself times yields , then the characteristic of the field is the least such . Otherwise, is defined to be .
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if is the prime subfield of , then is the size of if this is finite, and otherwise.
Note that the first definition also applies to arbitrary rings, and not just to fields.
The characteristic of a field (or more generally an integral domain) is always prime. For if the characteristic of were composite, say for , then in particular would equal zero. Then either would be zero or would be zero, so the characteristic of would actually be smaller than , contradicting the minimality condition.
Title | characteristic |
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Canonical name | Characteristic |
Date of creation | 2013-03-22 12:05:01 |
Last modified on | 2013-03-22 12:05:01 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 16 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 12E99 |