Monogenic field - Alchetron, The Free Social Encyclopedia

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 polynomial ring Z[a]. The powers of such an 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 = Q ( d ) with d a square-free integer, then O K = Z [ a ] where a = ( 1 + d ) / 2 if d ≡ 1 (mod 4) and a = d if d ≡ 2 or 3 (mod 4).

Cyclotomic fields:

if K = Q ( ζ ) with ζ a root of unity, then O K = Z [ ζ ] . Also the maximal real subfield Q ( ζ ) + = Q ( ζ + ζ − 1 ) is monogenic, with ring of integers Z [ ζ + ζ − 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

Monogenic field Wikipedia

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