Monogenic field

In mathematics, a monogenic field is an algebraic number field for which there exists an element a such that the ring of integers O_K 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 $K=\mathbf {Q} ({\sqrt {d}})$ 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 $O_{K}=\mathbf {Z} [\zeta ]$.

Not all number fields are monogenic: Dirichlet gave the example of the cubic field generated by a root of the polynomial $X^{3}-X^{2}-2X-8$.

References

• Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, 64. ISBN 3540219021.