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 subring Z[a] of K generated by a. Then OK 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:
if K = Q ( d ) {\displaystyle K=\mathbf {Q} ({\sqrt {d}})} with d {\displaystyle d} a square-free integer, then O K = Z [ a ] {\displaystyle O_{K}=\mathbf {Z} [a]} where a = ( 1 + d ) / 2 {\displaystyle a=(1+{\sqrt {d}})/2} if d ≡ 1 (mod 4) and a = d {\displaystyle a={\sqrt {d}}} if d ≡ 2 or 3 (mod 4). if K = Q ( ζ ) {\displaystyle K=\mathbf {Q} (\zeta )} with ζ {\displaystyle \zeta } a root of unity, then O K = Z [ ζ ] . {\displaystyle O_{K}=\mathbf {Z} [\zeta ].} Also the maximal real subfield Q ( ζ ) + = Q ( ζ + ζ − 1 ) {\displaystyle \mathbf {Q} (\zeta )^{+}=\mathbf {Q} (\zeta +\zeta ^{-1})} is monogenic, with ring of integers Z [ ζ + ζ − 1 ] {\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 X 3 − X 2 − 2 X − 8 {\displaystyle X^{3}-X^{2}-2X-8} , due to Richard 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.