cyclotomic extension

Take the vertices of a regular polygon drawn on the unit circle in the complex plane — these are the roots of unity, the solutions of x^n = 1. A cyclotomic extension is what you get by adjoining all of them to a base field. Because multiplying roots of unity just rotates around the same circle, the resulting symmetry is unusually tidy: the Galois group ends up abelian.

Concretely, the n-th cyclotomic extension of K is K(zeta_n), where zeta_n is a primitive n-th root of unity. Over the rationals, the minimal polynomial of zeta_n is the n-th cyclotomic polynomial Phi_n(x), which is irreducible over Q of degree phi(n) (Euler's totient). Hence [Q(zeta_n) : Q] = phi(n), and the extension is Galois because the conjugates of zeta_n (the other primitive n-th roots) all lie in Q(zeta_n).

The Galois group Gal(Q(zeta_n)/Q) is canonically isomorphic to the unit group (Z/nZ)^*: the automorphism sigma_a sends zeta_n to zeta_n^a for each a coprime to n, and sigma_a composed with sigma_b is sigma_{ab}. This makes the group abelian and ties number theory tightly to Galois theory — for instance the Kronecker-Weber theorem says every finite abelian extension of Q sits inside some cyclotomic field.