Monogenic field

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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 $O_K = \mathbf{Z}[\zeta]$ .