Constructing a vector space of dimension $2$ for a cyclotomic field

Question

Suppose that $z_n$ is a primitive $n$th root of unity. I want to find some subfield $F$ of $\mathbb{Q}(z_n)$ with $F \subseteq \mathbb{R}$ satisfying $\mbox{dim}_F \mathbb \,{\mathbb{Q}(z_n)} = 2$ (i.e., the vector space of $\mathbb{Q}(z_n)$ over $F$ has dimension $2$). Is there any way I can go about doing this without going too deep into Galois theory?

Answer 1

Hint: try $F=\mathbb{Q}(z_n+\overline{z_n})$.