cyclotomic field

Take the points evenly spaced around the unit circle in the complex plane — the n-th roots of unity, the corners of a regular n-gon — and adjoin them to the rationals. The “circle-dividing” field you obtain is among the most beautiful number fields known. Its symmetries are perfectly transparent, every abelian extension of Q hides inside one of them, and it is the natural home of cyclotomy and reciprocity.

The n-th cyclotomic field is K = Q(z_n), where z_n is a primitive n-th root of unity. Its degree over Q is phi(n), Euler's totient, and its ring of integers is exactly Z[z_n] — pleasantly, the naive guess is correct here. The minimal polynomial of z_n is the n-th cyclotomic polynomial, an irreducible monic integer polynomial of degree phi(n).

Cyclotomic fields are abelian Galois extensions of Q with Galois group isomorphic to (Z/nZ)^*, the unit group modulo n, via the map sending an automorphism to the exponent it applies to z_n. The Kronecker-Weber theorem says every abelian extension of Q lies inside some cyclotomic field. A prime p is ramified exactly when p divides n, and unramified primes split according to the order of p modulo n.