Monogenic field

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 $$K = \mathbf{Q}(\sqrt d)$$ with $$d$$ a square-free integer, then $$O_K = \mathbf{Z}[a]$$ where $$a = (1+\sqrt d)/2$$ if d ≡ 1 (mod 4) and $$a = \sqrt d$$ if d ≡ 2 or 3 (mod 4).


 * Cyclotomic fields:
 * if $$K = \mathbf{Q}(\zeta)$$ with $$\zeta$$ a root of unity, then $$O_K = \mathbf{Z}[\zeta].$$ Also the maximal real subfield $$\mathbf{Q}(\zeta)^{+} = \mathbf{Q}(\zeta + \zeta^{-1})$$ is monogenic, with ring of integers $$\mathbf{Z}[\zeta+\zeta^{-1}]$$.

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 $$X^3 - X^2 - 2X - 8$$, due to Richard Dedekind.