Cyclotomic extension

Summary

A cyclotomic extension of a field $K$ is obtained by adjoining roots of unity to $K$. The extension $K(\zeta_n)/K$ generated by a primitive $n$-th root of unity is Galois, and its Galois group embeds in $(\Z/n\Z)^\times$, so it is abelian. Over $\Q$ the embedding is an isomorphism, because the cyclotomic polynomials are irreducible over the rationals. The fields $\Q(\zeta_n)$ are the cyclotomic fields of number theory.