Consider the polynomial ring $GF (2)[x]/(x^n − 1)$. Let $f(x)$, $g(x)$ be two fixed polynomials in $GF(2)[x]/(x^n − 1)$.
Consider the subset generated by these two fixed polynomials: $⟨f (x), g(x)⟩ = {a(x)f (x) + b(x)g(x) | a(x), b(x) ∈ GF (2)[x]/(x^n − 1)}$. Using the definition of an ideal, prove that this subset is an ideal of $GF (2)[x]/(x^n − 1)$.
An ideal is by definition a set that is closed under addition and multiplication from the ring (or shorter, slicker: it is a $R$-submodule).
You just have to check that given two elements of the form $af+bg$ and $a^\prime fb^\prime g$ (suppressing the $x$ from the notation), then their sum is also of this form. This is easy.
Similarly, you have to check that the product of $af+bg$ is still of the same form (but with different $a,b$).
Hint: $h \cdot (af+bg) = haf+hbg$.