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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'''.
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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}}

Latest revision as of 06:03, 9 January 2015

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