View source for Monogenic field

{{TEX|done}}

{{MSC|12F}}

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 [[Discriminant of an algebraic number field|field discriminant]] of $K$ is equal to the [[discriminant of a polynomial|discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of $\alpha$.

Examples of monogenic fields include:
* [[Quadratic field]]s: 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 field]]s: 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