Strange outcomes for ring of integers in Sage

Question

The number field is $K=\mathbb{Q}(\alpha)=\mathbb{Q}[x]/(f)$ with $\alpha=\sqrt[4]{24}$ and $f=x^4-24$. Now Sage tells me that $\Delta(f)=-2^{17}\cdot3^3$ and $\Delta_K:=\Delta(\mathscr{O}_K)=-2^{11}\cdot3^3$. By the formula $\Delta(f)=\left[\mathscr{O}_K:\mathbb{Z}[\alpha]\right]^2\Delta_K$ we get that $\left[\mathscr{O}_K:\mathbb{Z}[\alpha]\right]=8$.

However, if I ask Sage to compute the ring of integers of $K$, it returns that it's the maximal order in $\alpha$ with defining polynomial $f$, which means that $\mathscr{O}_K=\mathbb{Z}[\alpha]$ which contradicts that index which is $8$. What went wrong? Here's my code:

K.<a>=NumberField(x^4-24)      #define numberfield
K.polynomial().discriminant()  #compute disc(f)
K.discriminant()               #discriminant of ring of integers (or equivalently the field)
K.ring_of_integers()           #unexpected outcome for ring of integers

Answer 1

You misinterpret the output of K.ring_of_integers(). It says:

Maximal Order in Number Field in a with defining polynomial x^4 - 24

It means:

Maximal Order in (Number Field in a with defining polynomial x^4 - 24)

or:

Maximal Order in K

It is just a name for the ring of integers; it does not give details about its structure.

What you probably wanted is a $\mathbb{Z}$-basis:

sage: OK = K.ring_of_integers()
sage: OK.basis()
[1, a, 1/2*a^2, 1/4*a^3]

Indeed, it is not $\mathbb{Z}[\alpha]$. You are right about the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_K$:

sage: O = K.order([a])
sage: O.index_in(OK)
8