characteristic

Field Characteristic

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For a field  with multiplicative identity 1, consider the numbers , , , etc. Either these numbers are all different, in which case we say that  has characteristic 0, or two of them will be equal. In the latter case, it is straightforward to show that, for some number , we have . If  is chosen to be as small as possible, then  will be a prime, and we say that  has characteristic . The characteristic of a field  is sometimes denoted .

The fields  (rationals),  (reals),  (complex numbers), and the p-adic numbers  have characteristic 0. For  a prime, the finite field GF() has characteristic .

If  is a subfield of , then  and  have the same characteristic.

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SEE ALSO

Characteristic, Field, Field Characteristic Exponent, Finite Field, Subfield