Solution
= Solution
{c}
<Polynomial division> gives a unique remainder of degree less than $n$ in every <coset>. Thus the classes
$$
1,[x],\ldots,[x]^{n-1}
$$
span the <quotient> as an $F$-vector space. A <linear combination> of them that vanishes would be a <polynomial> of degree less than $n$ divisible by $f$, so it is the zero <polynomial>. They are therefore <linearly independent>, proving
$$
\boxed{[K:F]=n.}
$$