Find the subfield diagram of splitting field of $x^{33}-1$ over $\mathbb{Q}$
This problem seems hard. I cannot find the fields using the automorphisms, because the basis is too large and so it would be too many computation. I know the galaois group is $Z_{10} \times Z_2$ thus we have $4$ non trivial subfields. I think I found $3$ $\mathbb{Q}(z_{11})$,$\mathbb{Q}(z_{3})$,$\mathbb{Q}(z_{33}+z_{33}^{-1})$ However, i am not sure how to find the inclusions, and what is the last subfield i ma missing. This is a problem on an algebra qual so should be doable using standard Galois theory.