 Finite field extension

A finite extension of the finite field ${\mathbb{F}}_q$ of degree $n$ is a field with $q^n$ elements. As a vector space, it has $n$ coordinates with $q$ choices each. Up to an isomorphism fixing ${\mathbb{F}}_q$, there is exactly one such field, denoted ${\mathbb{F}}_{q^n}$.

For example, ${\mathbb{F}}_4={\mathbb{F}}_2[t]/(t^2+t+1)$ has four classes $0,1,t,1+t$. It is not the quotient ring $\mathbb{Z}/4\mathbb{Z}$, which has a nonzero zero divisor. Its Galois group over ${\mathbb{F}}_q$ is generated by $x\mapsto x^q$. The phrase here means an extension of a finite coefficient field, not an arbitrary finite-degree extension of any field.