I am trying to prove that $f = X^3+2$ is irreducible in $\mathbf{F}_{49}[X]$, and whether $f$ is irreducible over all $\mathbf{F}_{7^n}$ for $n$ even.
Firstly I noticed that $\mathbf{F}_{49}[X] = \mathbf{F}_{7^2}[X]$. Now I have proven that $f$ is irreducible in $\mathbf{F}_{7}[X]$, by verifying that it has no roots in $\mathbf{F}_{7}$ by checking the 7 elements by hand. Does this say anything about whether it is irreducible in $\mathbf{F}_{7^2}[X]$? Is there any general strategy or approach to prove this, without checking all 49 elements for roots?
Since $X^3+2$ is irreducible over $\Bbb F_7$, any extension of $\Bbb F_7$ where it is reducible must contain $\Bbb F_{7^3}$, as that's the smallest extension of $\Bbb F_7$ where an irreducible cubic has a root.
Thus $\Bbb F_{49}$ cannot have a root. On the other hand, any extension of $\Bbb F_{7^3}$ will have a root. Thus, over, for instance, $\Bbb F_{7^6}=\Bbb F_{(7^3)^2}$ the polynomial will have a root and thus be reducible.