Primitive element of a finite field ($\mathrm{ord}(\alpha )=q-1$)

A primitive element of ${\mathbb{F}}_q$ generates its multiplicative group. This implies ${\mathbb{F}}_q={\mathbb{F}}_p(\alpha )$, but an element generating the field extension need not generate the multiplicative group. For instance $u\in {\mathbb{F}}_3[u]/(u^2+1)$ generates the quadratic extension but has order four, whereas $1+u$ has order eight and is multiplicatively primitive.

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.