= Primitive polynomial over a finite field
{title2=$\operatorname{ord}(\alpha)=q^m-1$}
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>.
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