How can i know if a field is monigenic?
I know the next field is monogenic but SageMath doesn't give me a power basis.
sage: c = sqrt(-(5+2*sqrt(5)))
sage: L.<c> = QQ[c]
sage: OL = L.maximal_order()
sage: B = OL.basis()
sage: L
Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: OL
Maximal Order in Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: B
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]
sage: L.integral_basis()
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]
The method from section 5 of Computing power integral bases in algebraic number fields yields that the powers of
(c + 1)/2form a basis ofOL. I don't know if there is a general method for fields of arbitrary degree, so are you interested specifically in quartic fields?