Although there can be multiple primitive root for a prime number but we are only concerned for smallest one.If you want to find all roots then continue the process till p-1 instead of breaking up on finding first primitive root. Proof It follows from the polar form of the unit roots. Primitive roots modulo a prime number were introduced by L. Euler, but the existence of primitive roots modulo an arbitrary prime number was demonstrated by C.F. Properties. Then a primitive root mod nexists if and only if n= 2, n= 4, n= pk or n= 2pk, where pis an odd prime. Theorem 3.5 (Primitive Roots Modulo Non-Primes) A primitive root modulo nis an integer gwith gcd(g;n) = 1 such that ghas order ˚(n). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … After looking at the properties of the cube roots of unity, we are ready to study the general properties of the nth roots of unity. 4- If it is 1 then 'i' is not a primitive root of n. 5- If it is never 1 then return i;. Once you have found one primitive root, you can easily find all the others. Property 2 The product of two unit roots is also a unit root. Title: proof of properties of primitive roots: Canonical name: ProofOfPropertiesOfPrimitiveRoots: Date of creation: 2013-03-22 18:43:48: Last modified on Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Property 1 The nth roots of unity have a unit modulus, that is: | | =. Gauss (1801). But if you are looking for primitive roots of, say, $2311$ then the probability of finding one at random is about 20% and there are 5 powers to test. How you find all the other primitive roots. A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.. References  S. Lang, "Algebra" , Addison-Wesley (1984)  Exercise 3.6. share | cite | improve this answer | follow | edited Oct 25 '16 at 22:17 (In general there are plenty of quadratic nonresidues that are not primitive roots, but this is easy to demonstrate).