Monogenic field

2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

An algebraic number field $K$ for which there exists an element $\alpha \in K$ such that the ring of integers $O_K$ is a polynomial ring $\mathbb{Z}[\alpha]$. The powers of such a element $\alpha$ 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 $\alpha$.

Examples of monogenic fields include:

• Quadratic fields: if $K = \mathbf{Q}(\sqrt d)$ with $d$ a square-free integer then $O_K = \mathbf{Z}[\alpha]$ where $\alpha = (1+\sqrt d)/2$ if $d \equiv 1 \pmod 4$ and $\alpha = \sqrt d$ otherwise.
• 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 $X^3 - X^2 - 2X - 8$.

References

• Narkiewicz, Władysław. Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag (2004), pp. 64, ISBN 3540219021