mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a monogenic field is an

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

''K'' for which there exists an element ''a'' such that the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

''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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...

of the minimal polynomial of α.

Examples

Examples of monogenic fields include: *

Quadratic fields In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

: : if

K = \mathbf(\sqrt d)

with

d

a square-free integer, then

O_K = \mathbf /math> 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(\zeta) with \zeta a root of unity, then O_K = \mathbf zeta Also the maximal real subfield \mathbf(\zeta)^ = \mathbf(\zeta + \zeta^) is monogenic, with ring of integers \mathbf zeta+\zeta^/math>.

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 - 2X - 8, due to

Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

References

* * Algebraic number theory {{Numtheory-stub