# What is a primitive polynomial?

What is a primitive polynomial? I was checking into some arbitrary number generation formulas and also 'primitive polynomial' showed up an enough variety of times that I determined to check into it in even more information.

I'm unclear of what a primitive polynomial is, and also why it serves for these arbitrary number generators.

I would certainly locate it specifically handy if an instance of a primitive polynomial can be given.

Consider a limited area $F_p$, after that we understand that it is cyclic. We call a component primitive if it creates this area. Better, offered an area and also some polynomial over that area (all the coefficients remain in the area), we can create an area expansion by any one of its origins. This is adjacent on that particular origin and also making an area of it.

It is a straightforward outcome of Galois Theory that if we take an area and also expand by some origin of some polynomial and also get a limited expansion (one that's level as a vector room over the initial area is limited), that we can locate a polynomial $m$ over our ground area such that $m$ disappears at this origin and also is marginal (tiniest level, i.e. it separates all various other polys which disappear at this origin).

If we take into consideration a primitive component and also its marginal polynomial, that poly is call primitive.

even more information on wiki

BBischof's solution is proper, yet however there's an additional, fairly various feasible definition of the very same term : that is a polynomial whose coefficients have no usual prime variable (this makes good sense over the integers, or various other UFDs too).