Let $K$ be the splitting field of $f=x^4+x^3+1$ over $\mathbb{F}_2$. Determine $K$ up to isomorphism, if $\alpha$ is a root of $f$ then describe explicitly all roots of $f$, and finally, find the least $m$ such that $f$ divides $x^m-1$ in $\mathbb{F}_2[x]$.
3 questions: I can show $f$ is irreducible over $\mathbb{F}_2$. So adjoining one root of $f$ to $\mathbb{F}_2$ generates a degree $4$ extension, namely $\mathbb{F}_{16}$. Since finite extensions of finite fields are normal, every other root of $f$ belongs to $\mathbb{F}_{16}$. Thus $K\cong\mathbb{F}_{16}$. Is this line of reasoning correct? Is it always the case that irreducible polynomials of degree $n$ over a finite field $\mathbb{F}_{p^k}$ will have splitting field $\mathbb{F}_{p^{nk}}$? Furthermore, I am a bit confused about the right approach for the second part. Should I represent $\mathbb{F}_{16}$ as the quotient $$ \mathbb{F}_{16}\cong\mathbb{F}_2[x]/(x^4+x^3+1) $$ and proceed from here? Finally, let's say I wanted to determine the least $m$ such that $f$ divides $x^m-1$ in $\mathbb{F}_2[x]$. Does this have to do with roots of unity? I'm wondering if this is related to the fact that $x^{p^n}-x=x^2-x$ is the product of every irreducible polynomial in $\mathbb{F}_2[x]$.
EDIT: After some discussion in the comments, my main question has been reduced to how we can find the least $m$ such that $f$ divides $x^m-1$ in $\mathbb{F}_2[x]$.