Symmetric convolution

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 $K=𝐐(\sqrt{d})$ 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 $O_K=𝐙[\zeta ].$

Not all number fields are monogenic; Richard Dedekind gave the example of the cubic field generated by a root of the polynomial $X^3-X^2-2X-8.$

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534