# Monogenic field

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In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the polynomial ring Z[a]. The powers of such an 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:

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).
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.