What is the basis for $[\mathbb{Q}(\gamma) : \mathbb{Q}]$?

Question

Let $\gamma = e^{2 \pi i/5} + (e^{2 \pi i/5})^4 %γ = e^{2πi / 5} + (e^{2πi / 5}) ⁴ $.

I am looking for the basis for $[\mathbb{Q}(\gamma):\mathbb{Q}] = 2$, and then looking for a dependence between $\gamma^2,\gamma$, and $1$.

I've worked all of this out by numerically but I am not sure how to do this through the basis.

Answer 1

Let $\omega = e^{2 \pi i/5}$ so that $\gamma = \omega + \omega^4$. We note that $$ \gamma^2 =\\ (\omega + \omega^4)^2 = \\ \omega^2 + 2\omega^5 + \omega^8 = \\ \omega^2 + 2 + \omega^3 = \\ 1 + \underbrace{(1 + \omega + \omega^2 + \omega^3 + \omega^4)}_{\text{what does this come out to?}} - \gamma $$

Answer 2

A series of hints: let the suggestively named $\zeta=e^{2\pi i/5}$, so that your $\gamma=\zeta+\zeta^4$.

• What is $\zeta+\zeta^2+\zeta^3+\zeta^4$? (Hint: $\frac{\zeta^5-1}{\zeta-1}=\ldots$)
• What is $\gamma^2$?
• What is $\gamma^2+\gamma$?

(This is also enough to give you an explicit expression for $\gamma$ in radicals, but you don't need that to solve the problem...)