Monogenic field

In mathematics, a monogenic field is an algebraic number field for which there exists an element a such that the ring of integers OK is a polynomial ring Z[a]. The powers of such a element a constitute a power integral basis.

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 $$O_K = \mathbf{Z}[\zeta]$$.

Not all number fields are monogenic: Dirichlet gave the example of the cubic field generated by a root of the polynomial $$X^3 - X^2 - 2X - 8$$.