Subfield fixed points and corresponding rational field
Let $F$ be the fixed field of $\{1,\tau_1\circ\tau_2\}$. Then $L/F$ is a Galois extension dimension 2. Clearly, $F\supseteq\Bbb Q(\sqrt[4]5(1+i))$ and the minimal polynomial of $\sqrt[4]5(1+i)$ over $\Bbb Q$ is $x^4+20$ which is irreducible over $\Bbb Q$ by Eiseinstein's criterion (for $p=5$). $$\underbrace{\Bbb Q\xrightarrow 4\Bbb Q(\sqrt[4]5(1+i))\to F\xrightarrow 2 L}_8$$ Thus $[\Bbb Q(\sqrt[4]5(1+i)):\Bbb Q]=4$, hence $F=\Bbb Q(\sqrt[4]5(1+i))$.