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}?

The degree $[\mathbb{Q}(\varepsilon):\mathbb{Q}]$ is precisely the dimension of $\mathbb{Q}(\varepsilon)$ as a $\mathbb{Q}$-vector space. — Clive Newstead, Mar 05 '15 at 18:58

score 3 · Accepted Answer · answered Mar 05 '15 at 18:57

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 $\left[ \mathbb{Q}(\varepsilon): \mathbb{Q} \right]=4$.

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

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