# Revision history [back]

### Hi, What does it mean that the relative discriminant is generated by a unit?, Does it mean that there is no relative integral basis?

sage: alpha=sqrt(-17/2 + sqrt(17^2-4(17))/2)
sage: f=alpha.minpoly()
sage: K.<a>=NumberField(x^4+17*x^2+17)
sage: R<y>=PolynomialRing(K)
sage: R(f).factor()
(y-a)(y+a)(y^2+a^2+17)

sage: g=y^2+a^2+17
sage: L.<b>=K.extension(g)
sage: L.relative_discriminant()
Fractional ideal (1)

 2 None Iguananaut 1708 ●8 ●33 ●55

### Hi, What does it mean that the relative discriminant is generated by a unit?, Does it mean that there is no relative integral basis?

sage: alpha=sqrt(-17/2 + sqrt(17^2-4(17))/2)
sage: f=alpha.minpoly()
sage: K.<a>=NumberField(x^4+17*x^2+17)
sage: R<y>=PolynomialRing(K)
sage: R(f).factor()
(y-a)(y+a)(y^2+a^2+17)

sage: g=y^2+a^2+17
sage: L.<b>=K.extension(g)
sage: L.relative_discriminant()
Fractional ideal (1)

 3 None Iguananaut 1708 ●8 ●33 ●55

### Hi, What does it mean that the relative discriminant is generated by a unit?, unit?

Does it mean that there is no relative integral basis?

sage: alpha=sqrt(-17/2 + sqrt(17^2-4(17))/2)
sage: f=alpha.minpoly()
sage: K.<a>=NumberField(x^4+17*x^2+17)
sage: R<y>=PolynomialRing(K)
sage: R(f).factor()
(y-a)(y+a)(y^2+a^2+17)

sage: g=y^2+a^2+17
sage: L.<b>=K.extension(g)
sage: L.relative_discriminant()
Fractional ideal (1)