I'll writezfor the primitive 7-th root of unity. The Galois group is acting by and is abstractly isomorphic to a cyclic group of order 6.

The subgroups of G are of order 2 and 3, generated by -1 and 2 say.

For the subgroup of order 3, consider . This is invariant under the action and so is in the fixed field. It has a conjugate which is obtained by applying . We observe that and on multiplying out, . We conclude that the subfield is .

For the subgroup of order 2, consider . This has conjugates and . We have , and . This identifies the subfield as a cubic extension of the rationals. Of course you can save yourself time by seeing that .

Since 2 and 3 are coprime, the subgroups of orders 2 and 3 intersect in the trivial subgroup, so the corresponding fields generate the whole extension: that is, .