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.
Examples of monogenic fields include:
- if with a square-free integer, then where if d ≡ 1 (mod 4) and if d ≡ 2 or 3 (mod 4).
- if with a root of unity, then Also the maximal real subfield is monogenic, with ring of integers
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 , due to Richard Dedekind.
- Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers (3rd ed.). Springer-Verlag. p. 64. ISBN 3-540-21902-1. Zbl 1159.11039.
- Gaál, István (2002). Diophantine Equations and Power Integral Bases. Boston, MA: Birkhäuser Verlag. ISBN 978-0-8176-4271-6. Zbl 1016.11059.
|This number theory-related article is a stub. You can help Wikipedia by expanding it.|