Cyclotomic extension

Definition. Let $K$ be a field and let $\zeta_n$ be a primitive n-th root of unity in some extension field. The extension

is called a cyclotomic extension. It is a simple extension generated by a root of the cyclotomic polynomial $\Phi_n(x)$.

Over $K=\mathbb{Q}$, the cyclotomic field $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$, and its Galois group is abelian (in fact isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$).

See also. Galois group , splitting field .

Examples.

1. $\mathbb{Q}(\zeta_3)=\mathbb{Q}\!\left(\frac{-1+i\sqrt3}{2}\right)=\mathbb{Q}(\sqrt{-3})$, a quadratic extension.
2. $\mathbb{Q}(\zeta_4)=\mathbb{Q}(i)$, also quadratic.
3. $\mathbb{Q}(\zeta_5)/\mathbb{Q}$ has degree $4$ since $\deg \Phi_5=\varphi(5)=4$.