= Solution
The <polynomial> $X^3+X+1$ has no root in $\mathbb F_2$, so it is an <irreducible polynomial>. The <quotient ring> of $\mathbb F_2[X]$ by the <ideal> it generates is therefore a <field> with eight elements. The seven nonzero elements form a <group> of order seven, so each satisfies $x^7=1$ by <Lagrange's theorem>. Thus this <field> splits $X^7-1$. A root different from one has order seven; it cannot lie in $\mathbb F_2$ and generates this degree-three <field extension>, since three is prime and the <tower law> applies. Therefore
$$
\boxed{[K:\mathbb F_2]=3.}
$$
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