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). share | cite | improve this answer | follow | edited Oct 25 '16 at 22:17 Title: proof of properties of primitive roots: Canonical name: ProofOfPropertiesOfPrimitiveRoots: Date of creation: 2013-03-22 18:43:48: Last modified on 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. Gauss (1801). Exercise 3.6. 4- If it is 1 then 'i' is not a primitive root of n. 5- If it is never 1 then return i;. (In general there are plenty of quadratic nonresidues that are not primitive roots, but this is easy to demonstrate). Properties. Property 2 The product of two unit roots is also a unit root. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 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. Property 1 The nth roots of unity have a unit modulus, that is: | | =. 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.. 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. How you find all the other primitive roots. References [1] S. Lang, "Algebra" , Addison-Wesley (1984) [2] 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. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Once you have found one primitive root, you can easily find all the others. Proof It follows from the polar form of the unit roots. Then a primitive root mod nexists if and only if n= 2, n= 4, n= pk or n= 2pk, where pis an odd prime.