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Math Help - Field Characteristic

  1. #1
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    Field Characteristic

    How can I show that every finite field F has a characteristic of the field using its definition?

    Thank you for any help.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    How can I show that every finite field F has a characteristic of the field using its definition?

    Thank you for any help.
    I do not follow you.
    How can you show what?
    That every finite field has prime chatachteristic?

    Or are you asking to show that evert finite field has a charachteristic?
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    I need to show the latter, that every finite field has a characteristic.
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    I need to show the latter, that every finite field has a characteristic.
    I am sure you are familar with the rule that all we need to do is check whether,
    n\cdot 1=0 for some positive integer to show there exists a charachteristic.
    Now, the additive group of the field is a finite group because the field is finite. Thus, 1\in <F,+> has finite order. Thus, n\cdot 1 = 0 where 0 is the identity element in <F,+>. Thus, it must have a non-zero charachteristic.
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