Primitive polynomial over a finite field ($\mathrm{ord}(\alpha )=q^m-1$)

A monic polynomial of degree $m$ over ${\mathbb{F}}_q$ that is an irreducible polynomial is primitive in the finite-field sense when one of its roots generates the multiplicative group of a finite field ${\mathbb{F}}_{q^m}^{\times }$. All its roots are Frobenius conjugates and then have the same order $q^m-1$. This notion differs from a primitive polynomial over an integral domain, which concerns the common divisor of its coefficients. Primitive finite-field polynomials provide explicit field representations and primitive cyclic Reed-Solomon codes or BCH codes.