Root

Definition (Root)

Given a field $\mathbb{F}$ and a polynomial $f\in\mathbb{F}[x]$ , we define $\alpha\in\mathbb{F}$ as a root of $f$ if $f(\alpha)=0$

Definition (Generator)

A generator $g$ of a finite field $\mathbb{F}_{q}$ is an element of order $q-1$ ; equivalently the powers of $g$ run through all the elements of $\mathbb{F}^{*}_{q}$ .

Intuition

So this generator concept proves that our finite field is of a cyclic nature. It also, in laymans terms, is a number that when taking powers up until its order, generates the distinct values that construct the set $\mathbb{F}_{q}$ again. So we “generate” $\mathbb{F}_{q}$ .

Definition (Primitive root)

A generator $g$ of $\mathbb{F}_{p}^{*}$ is called a primitive root if $g$ generates $\mathbb{F}_{p}^{*}$ .

Intuition

This is the same as a generator but in a specific context. Here its used specifically in the context of the multiplicative group of the finite field.

Linked from

Cyclic Group

Root

Quadratic Residue