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

• Cyclotomic fields:

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

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 ${\displaystyle X^{3}-X^{2}-2X-8}$, 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.