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==

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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 .

Not all number fields are monogenic: Dirichlet gave the example of the cubic field generated by a root of the polynomial .

References