I want to find a primitive element for the field extension $\mathbb{F}_{125}/\mathbb{F}_5$.
I constructed $\mathbb{F}_{125}$ as $\mathbb{F}_5[X]/\langle X^3 + X + 1 \rangle$. Since the degree of the polynomial is $3$, there are no intermediate fields, so every element from $\mathbb{F}_{125}$ which is not in $\mathbb{F}_{5}$ should be a primitive element.
But how do I know which elements are in $\mathbb{F}_{5}$ and which are not? For example, is $X + \langle X^3+X+1 \rangle$ in $\mathbb{F}_{5}$?
Best Answer
The extension field $\mathbf{F}_{125}$ contains prime field as a subfield, and as a vector space over it has $\{1,\bar X, \bar X^2\}$ as its basis. This shows that $\bar X$ is not in the prime field.
When the degree of a field extension is a prime number any element that is not in the base field will be a primitive element, as you have guessed in this spacial case.
Another easy way to construct a cubic extension over any field of $p$ elements, (or for that matter any field) is to look at the function: $x\mapsto x^3$ which is a homomorphism of $\mathbf{F}_{p}^*$ to itself. Then for any $b$ not in the image the polynomial $X^3-b$ would be irreducible over $\mathbf{F}_{p}$. Of course this is practical only when $p$ is small and so listing them is possible.