# Revision history [back]

### How can i know if a field is monigenic?

I know the next field is monogenic but sagameth doesn't give me a power basis.

c=sqrt(-(5+2*sqrt(5))) L.<c>=QQ[c] OL=L.maximal_order() B=OL.basis() L;OL;B;L.integral_basis()

Number Field in a with defining polynomial x^4 + 10x^2 + 5 with a = 0.?e-18 + 3.077683537175254?I Maximal Order in Number Field in a with defining polynomial x^4 + 10x^2 + 5 with a = 0.?e-18 + 3.077683537175254?I [3/8a^3 + 3/8a^2 + 1/8a + 1/8, 3/4a^3 + 1/4a, 1/2a^3 + 1/2a^2, a^3] [3/8a^3 + 3/8a^2 + 1/8a + 1/8, 3/4a^3 + 1/4a, 1/2a^3 + 1/2a^2, a^3] 2 None

### How can i know if a field is monigenic?

I know the next field is monogenic but sagameth SageMath doesn't give me a power basis.

c=sqrt(-(5+2*sqrt(5))) L.<c>=QQ[c] OL=L.maximal_order() B=OL.basis() L;OL;B;L.integral_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 + 10x^2 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?I
3.077683537175254?*I
sage: OL
Maximal Order in Number Field in a with defining polynomial x^4 + 10x^2 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?I
[3/8a^3 3.077683537175254?*I
sage: B
[3/8*a^3 + 3/8a^2 3/8*a^2 + 1/8a 1/8*a + 1/8, 3/4a^3 3/4*a^3 + 1/4a, 1/2a^3 1/4*a, 1/2*a^3 + 1/2a^2, 1/2*a^2, a^3]
[3/8a^3 sage: L.integral_basis()
[3/8*a^3 + 3/8a^2 3/8*a^2 + 1/8a 1/8*a + 1/8, 3/4a^3 3/4*a^3 + 1/4a, 1/2a^3 1/4*a, 1/2*a^3 + 1/2a^2, a^3]1/2*a^2, a^3]