Let $I=(X^3+2)$ be the principal ideal of $\mathbb{F}_7[X]$ generated by $X^3+2$. Show that $X^3+2$ is irreducible in $\mathbb{F}_7[X]$.
Can someone give me the first step on how to do this please. Also what is $\mathbb{F}_7[X]$? Is it just a polynomial with maximal degree of $7$ with coefficients of $\mathbb{F}$? Also what is $\mathbb{F}_7$?
The notation $\mathbb{F}_7[X]$ means the ring of polynomials in one variable over $\mathbb{F}_7$ the field with $7$ elements; this is basically $\mathbb{Z}/7\mathbb{Z}$.
A polynomial of degree $3$ that is reducible has a factor of degree $1$ and thus a root.
Check that your polynomial has no root, by simply plugging in all seven values.