List all subfields of the cyclotomic field $\mathbb{Q}\left(\boldsymbol{\mu}_{20}\right)$ obtained by adjoining all 20 th roots of unity to $\mathbb{Q}$, and draw the lattice diagram of inclusions among them. Write all the subfields in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Briefly justify your answer.

[The description of the Galois group of cyclotomic fields and the fundamental theorem of Galois theory can be used freely without proof.]