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How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
We fix the primitive roots of unity of order $7,11,13$, and denote them by $$ \tag {*} \zeta_7,\zeta_ {11},\zeta_ {13}\ . $$ Now we want to take each primitive root of prime order from above to some power, then multiply them. When the number of primes is small, or at least fixed, the notations are simpler.
The primitive $n^ {th}$ roots of unity form basis over $\mathbb {Q ...
In some contexts, the word primitive is used to mean a polynomial whose coefficients are relatively prime. In other contexts the word primitive is used to mean a polynomial a root of which generates a field under discussion. A polynomial that is primitive in the second sense must be irreducible.
field theory - How can I prove a polynomial to be primitive ...
Checking per Gerry's suggestion, a quick spreadsheet for the 40 primitive roots mod 101 shows that twenty-six (26) of them are square-free and fourteen (14) of them are not. We are helped in this by the fact that 2 is the smallest primitive root mod 101, so taking powers of 2 with exponents coprime to 100 gives all forty of the primitive roots (reduced mod 101).
A character is non-primitive iff it is of the form $1_ {\gcd (n,k)=1} \psi (n)$ with $\psi$ a character $\bmod m$ coprime with $k$. A character $\bmod p^2$ can be primitive with conductor $p$.