Subfields of field of cyclotomic 5th root of 1

Question

What are all the subfields of the cyclotomic 5th root of 1.

My attempt:The galois group over $\Bbb{Q}$ is $(\Bbb{Z}/5\Bbb{Z})^*=\{1,a^1,a^2,a^3\}$.The only possible non trivial subgroup is the subgroup generated by $a^2$.So the only possible non trivial subfield is the fixed field of this subgroup.

Answer 1

The Galois group shows that the only non-trivial subextension is quadratic, and algebraic number theory shows that the quadratic subfield of $\Bbb Q\left(\zeta_p\right)$ is $\Bbb Q\left(\sqrt{p^\ast}\right)$ where $p^* = (-1)^{\frac{p-1}2} p$ for any odd prime $p$, so in this case you get $\Bbb Q\left(\sqrt5\right)$.

It is left as an exercise to verify that indeed $\sqrt5 \in \Bbb Q\left(\zeta_5\right)$.