a. Determine $[\Bbb Q(\zeta_7 + {\zeta_7}^{-1}):\Bbb Q]$ .
b. Is $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$ a Galois extension?
c. Is $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$ a radical extension?
My attempt :
a. Computed $\min_{\Bbb Q}{(\zeta_7 + {\zeta_7}^{-1})}=x^3+x^2-2x+1$ , thus $[\Bbb Q(\zeta_7 + {\zeta_7}^{-1}):\Bbb Q]=3$ .
b. $\rm {Gal}(\Bbb Q(\zeta_7)|\Bbb Q) \cong \Bbb Z/6\Bbb Z$ and $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})$ is an intermediate field, thus $\Bbb Q(\zeta_7)| \Bbb Q(\zeta_7 + {\zeta_7}^{-1})$ is Galois and $\rm{Gal}(\Bbb Q(\zeta_7)| \Bbb Q(\zeta_7 + {\zeta_7}^{-1})) \lhd Gal(\Bbb Q(\zeta_7)|\Bbb Q)$ since the latter is cyclic, thus $\Bbb Q(\zeta_7 + {\zeta_7}^{-1})|\Bbb Q$ is Galois.
c. Since, $[\Bbb Q(\zeta_7 + {\zeta_7}^{-1}):\Bbb Q]=3$ in order for it to be radical, we simply need $\alpha \in \Bbb Q(\zeta_7 + {\zeta_7}^{-1})$ such that $\alpha^3 \in \Bbb Q$ . How to deal with this part?
Are my arguments valid? Thanks in advance for help!
Best Answer
You don't need to compute the minimal polynomial of $\eta = \zeta_7 + {\zeta_7}^{-1}$. Just apply Galois theory : $\eta$ is fixed by complex conjugation, hence $\mathbf Q(\eta) = \mathbf Q(\zeta_7) \cap \mathbf R$, cyclic of degree $3$. Besides, if $\mathbf Q(\eta)$ were a radical extension, it would be necessarily obtained by adding a cubic root $\alpha$ of a rational, and the normality of $\mathbf Q(\eta)$ would imply that $\mathbf Q(\eta)=\mathbf Q(\alpha,\omega)$, where $\omega$ is a primitive cubic root of unity, hence $\mathbf Q(\eta)$ would have degree $6$: contradiction.