Monogenic field explained

In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers O_K is the subring Z[''a''] of K generated by a. Then O_K 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:

• Quadratic fields:

if

K = Q (\sqrt d) with d a square-free integer, then O _{K} = 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 = Q (\zeta) with \zeta a root of unity, then O _{K} = Z [\zeta]. Also the maximal real subfield Q (\zeta) ^{+} = Q (\zeta + \zeta ^{-1} ) is monogenic, with ring of integers 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 , due to Richard Dedekind.

References

• Book: Narkiewicz, Władysław . Elementary and Analytic Theory of Algebraic Numbers . . 2004 . 64 . 3-540-21902-1 . 3rd . 1159.11039.
• Book: Gaál, István . Diophantine Equations and Power Integral Bases . . 2002 . Boston, MA . 978-0-8176-4271-6 . 1016.11059 .