Give an example of field extensions $L_{1}, L_{2}$ of $F$ for which $[L_{1}L_{1}:F]<[L_{1}:F][L_{2}:F]$.
$\bf{Solution}.$ We have $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=[\mathbb{Q}(\sqrt[4]{18}:\mathbb{Q})] = 4$ (Eisenstein criterion in $x^{4}-2$ and $x^{4}-18$). Say $a = \sqrt[4]{2}, b=\sqrt[4]{18}$. We have $b$ satisfies the polynomial $x^{2} - 3\sqrt{2} = x^{2} - 3a^{2} \in \mathbb{Q}(\sqrt[4]{2})[x]$, so $[\mathbb{Q}(\sqrt[4]{2},\sqrt[4]{18}):\mathbb{Q}(\sqrt[4]{18})] \leq 2.$ Thus $$[\mathbb{Q}(\sqrt[4]{2},\sqrt[4]{18}):\mathbb{Q}] = [\mathbb{Q}(\sqrt[4]{2},\sqrt[4]{18}):\mathbb{Q}(\sqrt[4]{2})][\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]\leq 8 < 16 = [\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}][\mathbb{Q}(\sqrt[4]{18}:\mathbb{Q})].$$
This is correct?
*$L_{1}L_{2}$ is the composition