= Solution
{c}
First address the general splitting-field request. A <splitting field> is an extension in which the <polynomial> factors into linear factors and which is generated over the base <field> by its roots. For a degree-$m$ <polynomial>, adjoin a root of an <irreducible> factor. The <quotient> construction in the preceding parts gives an extension of degree at most $m$. Remove the resulting linear factor and repeat, now with a <polynomial> of degree $m-1$ over that extension. Induction and the <tower law> give
$$
[L:F]\leq m(m-1)\cdots1=m!.
$$
The resulting <field> is generated by the roots and is a <splitting field>. Repeated roots or inseparability can only reduce the required degrees; separability was not used.
For the specified cubic, put $a=\sqrt[3]2$ and $\omega=e^{2\pi i/3}$. Its <splitting field> is $\mathbb Q(a,\omega)$. <Eisenstein criterion> at two gives $[\mathbb Q(a):\mathbb Q]=3$. This cubic <field> is real and therefore does not contain the nonreal $\omega$, whose quadratic <polynomial> is $x^2+x+1$. Consequently
$$
\boxed{[\mathbb Q(a,\omega):\mathbb Q]=3\cdot2=6.}
$$
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