In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the subring Z[a] of K generated by a. Then OK is a quotient of the polynomial ring Z[X] and the powers of a constitute a power integral basis. In a monogenic field K, the field discriminant of K is equal to the discriminant of the minimal polynomial of α. Examples Examples of monogenic fields include: Quadratic fields: if with a square-free integer, then where if d ≡ 1 (mod 4) and if d ≡ 2 or 3 (mod 4). Cyclotomic fields: if with a root of unity, then Also the maximal real subfield is monogenic, with ring of integers . While all quadratic fields are monogenic, already among cubic fields there are many that are not monogenic. The first example of a non-monogenic number field that was found is the cubic field generated by a root of the polynomial , due to Richard Dedekind. References Algebraic number theory