Find the degree of the splitting field

Question

Find the degree of the splitting field

1) $x^4 + 1$

2) $x^3 - 1$

3) $x^6 - 1$

Work:

1) $\mathbb{Q}(\sqrt (2i))$ is the splitting field for $x^4 + 1$ and so the degree is $4.$

2) $\mathbb{Q}(\omega)$ where $\omega = e^{\frac{2 \pi i}{3}}$ is the splitting field for $x^3 - 1$ and so the degree is $2.$

3) $\mathbb{Q}(\sqrt (3i))$ is the splitting field for $x^6 - 1$ and so the degree is $4.$

Answer 1

Let $\zeta_n$ be a primitive $n$-th root of unit.

1. The splitting field of $x^4 + 1$ is $\mathbb Q(\zeta_8)$.

2. The splitting field of $x^3 - 1$ is $\mathbb Q(\zeta_3)$.

3. The splitting field of $x^6 - 1$ is $\mathbb Q(\zeta_6)$.

You just need to find the degree of the cyclotomic field $\mathbb Q(\zeta_n)$ over $\mathbb Q$.