Cyclotomic field

A cyclotomic field is an extension of the rational numbers generated by a root of unity. Its Galois group is closely tied to the units modulo the root’s order.

Root of unity: A complex number whose positive integer power equals 1. A primitive root of unity generates a cyclotomic field.
Euler's totient function: The function counting positive integers up to n that are relatively prime to n. The degree of the nth cyclotomic field is φ(n).
Straightedge and compass construction: A geometric construction using only an unmarked straightedge and a compass. Constructible regular polygons correspond to specific cyclotomic subfields.
Quadratic field: A degree-two extension of the rational numbers. Quadratic fields are simpler number fields, while cyclotomic fields can have higher degree and contain them.
Primitive root of unity: A root of unity whose order is exactly the specified positive integer. Its order determines which cyclotomic field is generated.
Galois group: The group of field automorphisms fixing the base field in a Galois extension. Automorphisms describe the symmetries of a cyclotomic field.
Constructible polygon: A polygon whose vertices can be constructed using straightedge and compass. The constructibility of regular polygons is decided through roots of unity.
Real subfield of a cyclotomic field: The subfield fixed by complex conjugation inside a cyclotomic field. It contrasts the full field’s complex roots with its maximal real portion.
Field extension: A field containing a smaller field as a subfield. Adjoining a root of unity creates an extension of the rational numbers.
Unit group modulo n: The multiplicative group of residue classes modulo n that are relatively prime to n. Each such residue class determines an automorphism by acting on roots of unity.

Linked from 19 pages