Power Integral Basis

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''_''K'' is the subring Z 'a''of ''K'' generated by ''a''. Then ''O''_''K'' is a quotient of the polynomial ring Z 'X''and the powers of ''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 $K = \mathbf(\sqrt d)$ with $d$ a square-free integer, then O_K = \mathbf /math> where $a = (1+\sqrt d)/2$ if ''d'' ≡ 1 (mod 4) and $a = \sqrt d$ if ''d'' ≡ 2 or 3 (mod 4).
* Cyclotomic fields:
: if $K = \mathbf(\zeta)$ with $\zeta$ a root of unity, then $O_K = \mathbf$zeta Also the maximal real subfield $\mathbf(\zeta)^ = \mathbf(\zeta + \zeta^)$ is monogenic, with ring of integers \mathbf zeta+\zeta^/math>.

While all quadratic fields are monogenic, already among cubic fields there are many that are not monogenic. The first example of a non-monogenic number field that was found is the cubic field generated by a root of the polynomial $X^3 - X^2 - 2X - 8$, due to Richard Dedekind.

References

*
*

Algebraic number theory

{{Numtheory-stub