Monogenic field: Difference between revisions
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''a'' such that the [[ring of integers]] ''O''<sub>''K''</sub> is a polynomial ring '''Z'''[''a'']. The powers of such a element ''a'' constitute a '''power integral basis'''. | ''a'' such that the [[ring of integers]] ''O''<sub>''K''</sub> is a polynomial ring '''Z'''[''a'']. The powers of such a 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. | In a monogenic field ''K'', the [[Discriminant of an algebraic number field|field discriminant]] of ''K'' is equal to the [[discriminant of a polynomial|discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of α. | ||
==Examples== | ==Examples== | ||
Examples of monogenic fields include: | Examples of monogenic fields include: | ||
* [[Quadratic | * [[Quadratic field]]s: 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 | * [[Cyclotomic field]]s: 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: Dirichlet gave the example of the [[cubic field]] generated by a root of the polynomial <math>X^3 - X^2 - 2X - 8</math>. | Not all number fields are monogenic: Dirichlet gave the example of the [[cubic field]] generated by a root of the polynomial <math>X^3 - X^2 - 2X - 8</math>. | ||
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==References== | ==References== | ||
* {{cite book | last = Narkiewicz | first = Władysław | title = Elementary and Analytic Theory of Algebraic Numbers | * {{cite book | last = Narkiewicz | first = Władysław | title = Elementary and Analytic Theory of Algebraic Numbers | ||
| publisher = [[Springer-Verlag]] | year = 2004 | pages = 64 | isbn = 3540219021}} | | publisher = [[Springer-Verlag]] | year = 2004 | pages = 64 | isbn = 3540219021}}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:00, 21 September 2024
In mathematics, a monogenic field is an algebraic number field for which there exists an element a such that the ring of integers OK is a polynomial ring Z[a]. The powers of such a 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 a square-free integer then where if d≡1 (mod 4) and if d≡2 or 3 (mod 4).
- Cyclotomic fields: if with a root of unity, then .
![{\displaystyle O_{K}=\mathbf {Z} [a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edc1c1eabdf584070d6304e234d5deca0f654cfe)
![{\displaystyle O_{K}=\mathbf {Z} [\zeta ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df7a45d6e5d18e26d58f2493ff3df9449c4807fe)
Not all number fields are monogenic: Dirichlet gave the example of the cubic field generated by a root of the polynomial .
References
- Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, 64. ISBN 3540219021.