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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]] ''O''<sub>''K''</sub> is the [[polynomial ring]] '''Z'''[''a'']. The powers of such an element ''a'' 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 the [[Minimal polynomial (field theory)|minimal polynomial]] of α. | |||
==Examples== | |||
Examples of monogenic fields include: | |||
* [[Quadratic fields]]: | |||
: if <math>K = \mathbf{Q}(\sqrt d)</math> with <math>d</math> a [[square-free integer]], then <math>O_K = \mathbf{Z}[a]</math> where <math>a = (1+\sqrt d)/2</math> if ''d''≡1 (mod 4) and <math>a = \sqrt d</math> if ''d'' ≡ 2 or 3 (mod 4). | |||
* [[Cyclotomic fields]]: | |||
: if <math>K = \mathbf{Q}(\zeta)</math> with <math>\zeta</math> a [[root of unity]], then <math>O_K = \mathbf{Z}[\zeta].</math> | |||
Not all number fields are monogenic; [[Richard Dedekind]] gave the example of the [[cubic field]] generated by a root of the polynomial <math>X^3 - X^2 - 2X - 8.</math> | |||
==References== | |||
* {{cite book | last = Narkiewicz | first = Władysław | authorlink = | title = Elementary and Analytic Theory of Algebraic Numbers | publisher = [[Springer-Verlag]] | date = 2004 | location = | page = 64 | isbn = 3-540-21902-1 | edition=3rd | zbl=1159.11039}} | |||
* {{cite book | last = Gaál | first = István | authorlink = | title = Diophantine Equations and Power Integral Bases | publisher = [[Birkhäuser Verlag]] | date = 2002 | location = Boston, MA | isbn = 978-0-8176-4271-6 | zbl=1016.11059 }} | |||
[[Category:Algebraic number theory]] | |||
{{Numtheory-stub}} | |||
Revision as of 01:08, 17 December 2013
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:
- 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
Not all number fields are monogenic; Richard Dedekind gave the example of the cubic field generated by a root of the polynomial
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534