Monogenic field

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 [math]\displaystyle{ K = \mathbf{Q}(\sqrt d) }[/math] with [math]\displaystyle{ d }[/math] a square-free integer, then [math]\displaystyle{ O_K = \mathbf{Z}[a] }[/math] where [math]\displaystyle{ a = (1+\sqrt d)/2 }[/math] if d ≡ 1 (mod 4) and [math]\displaystyle{ a = \sqrt d }[/math] if d ≡ 2 or 3 (mod 4).

• Cyclotomic fields:

if [math]\displaystyle{ K = \mathbf{Q}(\zeta) }[/math] with [math]\displaystyle{ \zeta }[/math] a root of unity, then [math]\displaystyle{ O_K = \mathbf{Z}[\zeta]. }[/math] Also the maximal real subfield [math]\displaystyle{ \mathbf{Q}(\zeta)^{+} = \mathbf{Q}(\zeta + \zeta^{-1}) }[/math] is monogenic, with ring of integers [math]\displaystyle{ \mathbf{Z}[\zeta+\zeta^{-1}] }[/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 [math]\displaystyle{ X^3 - X^2 - 2X - 8 }[/math], due to Richard Dedekind.

References

• Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers (3rd ed.). Springer-Verlag. p. 64. ISBN 3-540-21902-1. 
• Gaál, István (2002). Diophantine Equations and Power Integral Bases. Boston, MA: Birkhäuser Verlag. ISBN 978-0-8176-4271-6.