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:

• a square-free integer if d≡1(mod if d ≡ 2 or 3 (mod 4).

• Cyclotomic fields:

• a root of unity.

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, due to Richard Dedekind.

References