Inmathematics, amonogenic field is analgebraic number fieldK for which there exists an elementa such that thering of integersO_K is the subringZ[a] ofK generated bya. ThenO_K is a quotient of thepolynomial ringZ[X] and the powers ofa constitute apower integral basis.

In a monogenic fieldK, thefield discriminant ofK is equal to thediscriminant of theminimal polynomial of α.

Examples of monogenic fields include:

• Quadratic fields:

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

• Cyclotomic fields:

if${\displaystyle K=\mathbf{Q}(\zeta )}$ with${\displaystyle \zeta }$ aroot 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 thecubic field generated by a root of the polynomial${\displaystyle X^3-X^2-2X-8}$, due toRichard Dedekind.

• 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.

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