subfields of the cyclotomic fields $\mathbb{Q}(\zeta_p)$ with $p \equiv 1 [5]$

Question

Let $p\equiv 1 (mod 5)$ a prime number, let $\mathbb{Q}(\zeta_p)$ the cyclotomic field of degree $p-1$ over $\mathbb{Q}$, its known that $\mathbb{Q}(\zeta_p)$ has unique subfield of degree $5$. My question about the explicit determination of this subfield, is there any information ??

Answer 1

Let $g$ be a generator of $$\Bbb{Z/pZ}^\times\cong Gal(\Bbb{Q(\zeta_p)/Q})$$ The unique subfield such that $[K:\Bbb{Q}]=5$ is the unique subfield such that $Gal(\Bbb{Q}(\zeta_p)/K)$ is a subgroup of index $5$, thus it is the subfield fixed by every $$\sigma_{g^{5n}}(\zeta_p^a)= \zeta_p^{a g^{5n}}\in Gal(\Bbb{Q(\zeta_p)/Q})$$

Thus $$K = \Bbb{Q}( \{ \sum_{l=1}^{(p-1)/5} \sigma_{g^{5n}}(\zeta_p^a), a\le p-1\})=\Bbb{Q}(\sum_{l=1}^{(p-1)/5} \sigma_{g^{5n}}(\zeta_p))$$ where the last step is because $\Bbb{Q}(\sum_{n=1}^{(p-1)/5} \sigma_{g^{5n}}(\zeta_p))/\Bbb{Q}$ is abelian so it contains the conjugates of $\sum_{n=1}^{(p-1)/5} \sigma_{g^{5n}}(\zeta_p)$ already.