Integral Bases and Discriminants in Number Fields

The Discriminant of a Basis

We begin by defining the discriminant for an arbitrary $n$-tuple of elements in a number field $K$ of degree $n$. This definition builds directly on the concepts of trace and embeddings from our last lesson.

There are two equivalent and equally important ways to define this.

Definition 1 (via Embeddings): Let $K$ be a number field of degree $n$ with embeddings ${\sigma }_1,\dots ,{\sigma }_n:K\hookrightarrow \mathbb{C}$. For any elements ${\alpha }_1,\dots ,{\alpha }_n\in K$, their discriminant is:

where ${\sigma }_i({\alpha }_j)$ is the entry in the $i$-th row and $j$-th column of an $n\times n$ matrix. Squaring the determinant ensures the result is independent of how we order the embeddings or the elements.

Definition 2 (via Trace): The discriminant can also be defined using the trace bilinear form, $(x,y)\mapsto {\text{Tr}}_{K/\mathbb{Q}}(xy)$.

The following reading formalizes these definitions and establishes their key properties.

Integral Basis and the Field Discriminant

Just as a vector space has a basis, a ring of integers $O_K$ has a special kind of basis called an integral basis. Because $O_K$ is a free $\mathbb{Z}$-module of rank $n=[K:\mathbb{Q}]$, this means we can find $n$ elements that generate all of $O_K$ using only integer coefficients.

Definition: An integral basis for a number field $K$ is a set of elements $\{{\omega }_1,\dots ,{\omega }_n\}$ in $O_K$ such that every element $\alpha \in O_K$ can be written uniquely as a $\mathbb{Z}$-linear combination $\alpha =c_1{\omega }_1+\cdots +c_n{\omega }_n$ with $c_i\in \mathbb{Z}$.

It is a non-trivial theorem that every number field has an integral basis. With this concept, we can now define the discriminant of the field itself.

Definition: The discriminant of a number field K, denoted $d_K$ (or $D_K$), is the discriminant of any integral basis of $K$.

A natural question arises: does the choice of integral basis matter? Fortunately, it does not. If $\{{\omega }_1,\dots ,{\omega }_n\}$ and $\{{\eta }_1,\dots ,{\eta }_n\}$ are two integral bases, they are related by a change-of-basis matrix $A$ with integer entries. Since the transformation is invertible, $A^{-1}$ must also have integer entries, which implies that $\det (A)=\pm 1$. The relationship between the discriminants of two bases is given by $\text{disc}({\eta }_1,\dots ,{\eta }_n)=(\det A{)}^2\text{disc}({\omega }_1,\dots ,{\omega }_n)$. Since $(\det A{)}^2=(\pm 1{)}^2=1$, the discriminant is the same for any integral basis, making $d_K$ a well-defined invariant of the field.

Computing the Discriminant

Finding an integral basis and the discriminant can be challenging. The general strategy involves starting with a simple $\mathbb{Q}$-basis of algebraic integers and refining it. The most common starting point is a power basis.

For a number field $K=\mathbb{Q}(\alpha )$ of degree $n$, the set $\{1,\alpha ,\dots ,{\alpha }^{n-1}\}$ forms a $\mathbb{Q}$-basis for $K$. If $\alpha$ is an algebraic integer, this gives us a subring $\mathbb{Z}[\alpha ]\subseteq O_K$. The discriminant of this power basis can be computed relatively easily using the minimal polynomial of $\alpha$.

This gives us the discriminant of the subring $\mathbb{Z}[\alpha ]$, but not necessarily the discriminant of the full ring of integers $O_K$. The two are related by a fundamental formula.

The Index Formula:

Here, $[O_K:\mathbb{Z}[\alpha ]]$ is the index of the subgroup $\mathbb{Z}[\alpha ]$ in the group $O_K$ (under addition). It measures how "much bigger" $O_K$ is than $\mathbb{Z}[\alpha ]$. This formula is our primary tool. We compute the left side, which is easy. If the result is a square-free integer, then the index must be 1, which means $O_K=\mathbb{Z}[\alpha ]$ and the power basis is an integral basis. If it is not square-free, we must investigate integers dividing the index.

Examples of Computation

Let's apply this strategy to the most important cases.

Quadratic Fields: $K=\mathbb{Q}(\sqrt{d})$

Let $d$ be a square-free integer. We start with the simple basis $\{1,\sqrt{d}\}$ for $K$, where $\alpha =\sqrt{d}$. The minimal polynomial is $f(x)=x^2-d$.
The degree is $n=2$, so $n(n-1)/2=1$. The derivative is $f^{\prime }(x)=2x$.

Using the multiplicativity of the norm, this is $-N(2)N(\sqrt{d})=-(2^2)(d)=-4d$. Wait, $N(\sqrt{d})=(\sqrt{d})(-\sqrt{d})=-d$, so this should be $-(2^2)(-d)=4d$. Let's recompute using embeddings ${\sigma }_1(a+b\sqrt{d})=a+b\sqrt{d}$ and ${\sigma }_2(a+b\sqrt{d})=a-b\sqrt{d}$:

So, we have $\text{disc}(1,\sqrt{d})=4d$. From the index formula, $4d=[O_K:\mathbb{Z}[\sqrt{d}]{]}^2\cdot d_K$. This implies that the index can only be 1 or 2.

• Case 1: $d\equiv 2,3(\mathrm{mod}4)$
  It can be shown that there are no algebraic integers in $O_K$ that are not in $\mathbb{Z}[\sqrt{d}]$. Therefore, the index is 1, $\mathbb{Z}[\sqrt{d}]$ is the full ring of integers, $\{1,\sqrt{d}\}$ is an integral basis, and $d_K=4d$.

• Case 2: $d\equiv 1(\mathrm{mod}4)$
  In this case, consider the element $\omega =\frac{1+\sqrt{d}}{2}$. Its minimal polynomial is $x^2-x+\frac{1-d}{4}=0$. Since $d\equiv 1(\mathrm{mod}4)$, $1-d$ is divisible by 4, so this polynomial has integer coefficients. Thus, $\omega$ is an algebraic integer. This element is in $O_K$ but not in $\mathbb{Z}[\sqrt{d}]$. The ring of integers is actually $O_K=\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$. The integral basis is $\{1,\frac{1+\sqrt{d}}{2}\}$, and a similar calculation shows that $d_K=d$.

These crucial results are summarized here:

Cubic Field Example: $K=\mathbb{Q}(\theta )$ with ${\theta }^3-\theta +1=0$

Let $f(x)=x^3-x+1$. This polynomial is irreducible over $\mathbb{Q}$. Let's compute the discriminant of the power basis $\{1,\theta ,{\theta }^2\}$. Here $n=3$, so $n(n-1)/2=3$. The derivative is $f^{\prime }(x)=3x^2-1$.

To compute this norm, we need the minimal polynomial of $\gamma =3{\theta }^2-1$. This is a tedious calculation by hand. A more direct method uses resultants, but we can also follow the worked example provided in the resource.

As you saw, $\text{disc}(1,\theta ,{\theta }^2)=-23$. Since -23 is a square-free integer, the index $[O_K:\mathbb{Z}[\theta ]]$ must be 1. This means $O_K=\mathbb{Z}[\theta ]$, so $\{1,\theta ,{\theta }^2\}$ is an integral basis, and the field discriminant is $d_K=-23$.

Computation in PARI/GP

Finding integral bases and discriminants by hand quickly becomes difficult. PARI/GP automates this entire process with the nfinit command.

When you execute K = nfinit(poly), PARI/GP performs a complex algorithm (the Round-2 or Round-4 algorithm) to compute the fundamental invariants of the number field defined by poly, including the discriminant and an integral basis. The results are stored in a structure, which we've named K.

You can access the components of this structure with a dot (.).

• K.disc: The field discriminant $d_K$.
• K.zk: The integral basis, given as a vector of polynomials in the generator x.

Let's verify our previous examples.

1. Quadratic Field $\mathbb{Q}(\sqrt{13})$
Since $d=13\equiv 1(\mathrm{mod}4)$, we expect $d_K=13$ and an integral basis $\{1,(1+\sqrt{13})/2\}$.

? K = nfinit(x^2 - 13);
? K.disc
%1 = 13
? K.zk
%2 = [1, 1/2*x + 1/2]

The output matches our prediction. Note that x represents $\sqrt{13}$, so 1/2*x + 1/2 is $\frac{\sqrt{13}+1}{2}$.

2. Quadratic Field $\mathbb{Q}(\sqrt{10})$
Since $d=10\equiv 2(\mathrm{mod}4)$, we expect $d_K=4\cdot 10=40$ and an integral basis $\{1,\sqrt{10}\}$.

? K = nfinit(x^2 - 10);
? K.disc
%3 = 40
? K.zk
%4 = [1, x]

Again, the results match perfectly.

3. Cubic Field $\mathbb{Q}(\theta )$ with ${\theta }^3-\theta +1=0$
We found $d_K=-23$ and the integral basis to be $\{1,\theta ,{\theta }^2\}$.

? K = nfinit(x^3 - x + 1);
? K.disc
%5 = -23
? K.zk
%6 = [1, x, x^2]

PARI/GP confirms our hand calculation.

Conclusion

In this lesson, we have defined two of the most fundamental objects associated with a number field: the integral basis $\{{\omega }_i\}$ and the field discriminant $d_K$.

Key takeaways:

• The discriminant of a basis measures whether the elements are linearly independent and is computed using traces or embeddings.
• An integral basis is a $\mathbb{Z}$-basis for the ring of integers $O_K$.
• The discriminant of the field, $d_K$, is the discriminant of any integral basis. It is a well-defined integer invariant of the field.
• The index formula, $\text{disc}(f)=[O_K:\mathbb{Z}[\alpha ]{]}^2\cdot d_K$, is the essential tool for computing $d_K$ and determining if a power basis $\mathbb{Z}[\alpha ]$ is the full ring of integers $O_K$.
• The nfinit command in PARI/GP is the practical tool for finding $d_K$ and the integral basis for any given number field.

We have now assembled the basic toolkit for describing number fields. We know what a number field is, what its ring of integers is, and how to compute its most basic invariants. In our next lesson, we will complete this foundational module by exploring how to use the structures created by nfinit to perform arithmetic with elements and set the stage for our main topic: the arithmetic of ideals.