I am currently reading a coding theory book and I stumbled upon the following:
Suppose we have a cyclic code $C\subseteq\mathbb{F}_{q}^n$. If $\text{gcd}(q,n)=1$ we can find an $m$ such that $n|q^m-1$ and thus $x^n-1$ splits in $\mathbb{F}_{q^m}$. Following this, we can factorize $x^7-1$ in $\mathbb{F}_2[x]$ as $(x+1)(x^3+x+1)(x^3+x^2+1)$. Notice now that $x^7-1$ splits in $\mathbb{F}_{2^3}$ as $\prod_{i=0}^6 (x-\alpha^i)$ where $\alpha$ is a primitive $7$-th root of unity. We then look at the code generated by $g(x)=(x^3+ x^2+1)$. This factors into $(x-\alpha^3)(x-\alpha^5)(x-\alpha^6)$ in $\mathbb{F}_{2^3}$.
I don't understand what is meant by factors into in this context. Can anyone explain?