- 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 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 square-free 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 X3 − X2 − 2X − 8.
References
- Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag. pp. 64. ISBN 3540219021.
- Gaál, István (2002). Diophantine Equations and Power Integral Bases. Birkhäuser Verlag. ISBN 978-0-8176-4271-6.
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