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Old 2018-02-08, 05:15   #1
devarajkandadai's Avatar
May 2004

22·79 Posts
Default primitive roots- when the base is a quadratic algebraic integer

As members are aware we had to modify Fermat's theorem in order to accommodate quadratic algebraic integers as bases ( see thread: conjecture pertaining to modified Fermat's theorem ). The modified theorem:

a^(p^2-1) = = 1(mod(p)). This is subject to conditions mentioned in the thread.

Now naturally the concept of primitive roots in this context arises. We can now
say that a is a primitive root of m if a^(eulerphi(m^2)) = = 1 subject to
i) m is coprime with norm of a
ii) m is coprime with the discriminant
iii) a^k is not congruent to 1 (mod p) where k is any rational integer less than eulerphi(p^2-1)
Here, of course, a is a quadratic algebraic integer. In my next post will give examples.
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