Monogenic field
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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 subring Z[a] of K generated by a. Then OK is a quotient of the polynomial ring Z[X] and the powers of 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
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.
References
- 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.