Cyclotomic field

Summary

The $n$-th cyclotomic field is the field $\Q(\zeta_n)$ obtained by adjoining to $\Q$ a primitive $n$-th root of unity $\zeta_n$. It is a Galois extension of $\Q$ of degree $\varphi(n)$ whose Galois group is the group of units of $\Z/n\Z$. By the Kronecker–Weber theorem, every finite abelian extension of $\Q$ lies in some cyclotomic field. Cyclotomic fields were studied by Kummer in his work on Fermat's Last Theorem, which led to the notion of a regular prime.