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

Question

Let ɛ = e^{2πi / 5}.

$[\mathbb{Q}(ɛ) : \mathbb{Q}]$ = 4
I originally thought the dimension was 2 with basis {1, ɛ}, but it is actually 4. What exactly is the basis, and why is it not {1, ɛ}?

Answer 1

One basis of that extension is $\{ 1, \varepsilon, \varepsilon^2,\varepsilon^3 \}$. You can see $\mathbb{Q(\varepsilon)}$ as the splitting field of the 5th cyclotomic polinomial:

$$ \Phi_5(x)=x^4+x^3+x^2+x^1+1$$

Which is irreducible and $deg(\Phi_5)=4$, so $\left[ \mathbb{Q}(\varepsilon): \mathbb{Q} \right]=4$.