The integers mod p are a field for any prime p, so we can add, subtract, multiply and divide (except by 0) at will. Also, any polynomial equation of degree n has at most n solutions. Multiplying the nonzero elements together, we must get a number that is -1 mod p, because an even number of these elements are not self-reciprocal and therefore the reciprocals pair up, while the self-reciprocal element(s) are p-1 (congruent to -1) and 1 (since the equation x^2=1 has only these 2 solutions).
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Posted by Richard
on 2003-11-21 14:32:16 |
We are in a field
