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 $K=\mathbf {Q} ({\sqrt {d}})$ with $d$ a square-free integer, then $O_{K}=\mathbf {Z} [a]$ 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 $K=\mathbf {Q} (\zeta )$ with $\zeta$ a root of unity, then $O_{K}=\mathbf {Z} [\zeta ].$ Also the maximal real subfield $\mathbf {Q} (\zeta )^{+}=\mathbf {Q} (\zeta +\zeta ^{-1})$ is monogenic, with ring of integers $\mathbf {Z} [\zeta +\zeta ^{-1}]$.

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

• Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers (3rd ed.). Springer-Verlag. p. 64. ISBN 3-540-21902-1. Zbl 1159.11039.
• Gaál, István (2002). Diophantine Equations and Power Integral Bases. Boston, MA: Birkhäuser Verlag. ISBN 978-0-8176-4271-6. Zbl 1016.11059.