Monogenic field

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 polynomial ring Z[a]. The powers of such an element 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 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 ζ a root of unity, then O_K = \mathbf{Z}[\zeta].

Not all number fields are monogenic; Richard Dedekind gave the example of the cubic field generated by a root of the polynomial X3X2 − 2X − 8.


  • Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag. pp. 64. ISBN 3540219021. 
  • Gaál, István (2002). Diophantine Equations and Power Integral Bases. Birkhäuser Verlag. ISBN 978-0-8176-4271-6.