 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 O_{K} 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
Examples of monogenic fields include:
 Quadratic fields:
 if with d a squarefree integer, then where if d≡1 (mod 4) and if d ≡ 2 or 3 (mod 4).
 Cyclotomic fields:
 if with ζ a root of unity, then
Not all number fields are monogenic; Richard Dedekind gave the example of the cubic field generated by a root of the polynomial X^{3} − X^{2} − 2X − 8.
References
 Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers. SpringerVerlag. pp. 64. ISBN 3540219021.
 Gaál, István (2002). Diophantine Equations and Power Integral Bases. Birkhäuser Verlag. ISBN 9780817642716.
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