Let $K$ be a Galois extension of $F$ The extension is said to be an abelian extension if the Galois group ${Gal$(K/F)$}$ is abelian.

Examples: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ has Galois group $\mathbb{Z}/2\mathbb{Z}$ so $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is an abelian extension.

Let $\zeta_n$ be a primitive nth root of unity. Then $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ has Galois group $(\mathbb{Z}/n\mathbb{Z})^*$ (the group of units of $\mathbb{Z}/n\mathbb{Z}$ so $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is abelian.