9 What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more detail. I'm unsure of what a primitive polynomial is, and why it is useful for these random number generators.
A primitive root of a prime $p$ is an integer $g$ such that $g\ (\bmod\ p)$ has multiplicative order $p-1$ The main thing I'm confused about is what "multiplicative order" is.
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
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).
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 ...
I will really appreciate if someone could give example of these and explain what makes them primitive and prime. e.g about polynomials used in Cyclic Redundancy Check that are implemented using hardware feedback registers it says "The best ones are not necessarily prime (irreducible) nor primitive"