Number Fields and Rings of Integers

What is a Number Field?

In essence, algebraic number theory studies number systems that are larger than the rational numbers $\mathbb{Q}$ but still manageable. These systems are called number fields. Formally, a number field $K$ is a field that is a finite extension of the field of rational numbers $\mathbb{Q}$.

This means two things:

1. $K$ is a field containing $\mathbb{Q}$ as a subfield.
2. When viewed as a vector space over $\mathbb{Q}$, $K$ has a finite dimension. This dimension is called the degree of the extension, denoted $[K:\mathbb{Q}]$.

To fully unpack this, it's helpful to review a few foundational concepts from field theory. The following reading from Martin Orr's "Algebraic Number Theory" lecture notes provides a clear and efficient summary. Given your background in mathematics, this should serve as a good refresher.

The Tower Law is particularly important. For a tower of finite extensions $K\subset L\subset M$, it states that $[M:K]=[M:L][L:K]$. This multiplicative property of degrees is analogous to how dimensions of nested vector subspaces relate.

For instance, consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$, which is obtained by adjoining both $\sqrt{2}$ and $\sqrt{3}$ to $\mathbb{Q}$. We can view this as a tower of extensions: $\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})\subset \mathbb{Q}(\sqrt{2},\sqrt{3})$.

• We know $[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2$, since $\{1,\sqrt{2}\}$ is a basis.
• The minimal polynomial of $\sqrt{3}$ over $\mathbb{Q}$ is $x^2-3$. This polynomial has no roots in $\mathbb{Q}(\sqrt{2})$, so it remains the minimal polynomial of $\sqrt{3}$ over $\mathbb{Q}(\sqrt{2})$. Therefore, $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}(\sqrt{2})]=2$.

By the Tower Law, the total degree is $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=2\times 2=4$. This field contains intermediate subfields, which can be organized into a lattice diagram.

A key result, which we will use without proof, is the Primitive Element Theorem. It states that every number field $K$ can be generated by a single element, meaning $K=\mathbb{Q}(\alpha )$ for some algebraic number $\alpha$. This is convenient because it allows us to represent the entire field using polynomials in that single element $\alpha$.

The Ring of Integers

Now that we have defined our fields, we need to identify the "integers" within them. We can't simply take elements of the form $a_0+a_1\alpha +\dots$ where the coefficients $a_i$ are in $\mathbb{Z}$, because this definition would depend on our choice of the primitive element $\alpha$. For example, $\mathbb{Q}(\sqrt{2})=\mathbb{Q}(\sqrt{8})$, but the ring $\mathbb{Z}[\sqrt{2}]$ is different from $\mathbb{Z}[\sqrt{8}]$. We need a more intrinsic definition.

The correct approach, developed by Dedekind, is to define what it means for a number to be an "algebraic integer" and then to collect all such numbers that lie within our number field.

The following video gives an excellent historical and conceptual introduction to this idea.

As the video explains, an algebraic integer is a root of a monic polynomial with coefficients in $\mathbb{Z}$. For example, $\sqrt{2}$ is an algebraic integer because it's a root of the monic polynomial $x^2-2\in \mathbb{Z}[x]$. The golden ratio $\varphi =\frac{1+\sqrt{5}}{2}$ is an algebraic integer because it's a root of $x^2-x-1\in \mathbb{Z}[x]$. In contrast, $\frac{1}{2}$ is not, as its minimal polynomial over $\mathbb{Q}$ is $x-\frac{1}{2}$, which is not in $\mathbb{Z}[x]$.

This leads to a powerful criterion, which is one of the most useful characterizations of algebraic integers.

With these definitions, we can now formally define the central object of study:
The ring of integers of a number field $K$, denoted $O_K$, is the set of all elements in $K$ that are algebraic integers. That is, $O_K=K\cap \mathbb{A}$, where $\mathbb{A}$ is the ring of all algebraic integers.

For $K=\mathbb{Q}$, the ring of integers is just $O_{\mathbb{Q}}=\mathbb{Z}$. But for other number fields, it can be more interesting. The canonical example is the ring of integers of a quadratic field.

The result from the reading is worth summarizing: for $K=\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer:

• If $d\equiv 2,3(\mathrm{mod}4)$, then $O_K=\mathbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b\in \mathbb{Z}\}$.
• If $d\equiv 1(\mathrm{mod}4)$, then $O_K=\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right]=\left\{a+b\frac{1+\sqrt{d}}{2}\mid a,b\in \mathbb{Z}\right\}$.

This explains why for $K=\mathbb{Q}(i)=\mathbb{Q}(\sqrt{-1})$, where $d=-1\equiv 3(\mathrm{mod}4)$, the ring of integers is $\mathbb{Z}[i]$. However, for $K=\mathbb{Q}(\sqrt{5})$, where $d=5\equiv 1(\mathrm{mod}4)$, the ring of integers is the larger ring $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$. Here, $\mathbb{Z}[\sqrt{5}]$ is an order, but not the full ring of integers (which is the maximal order).

First Steps with PARI/GP

You expressed a preference for using PARI/GP for computations. Let's see how to define the objects we've just discussed.

You can start PARI/GP by typing gp in your terminal. We define a number field by specifying the minimal polynomial of a generator. For example, to define $K=\mathbb{Q}(\sqrt{5})$, whose generator $\sqrt{5}$ is a root of $x^2-5=0$, you would type:

K = nfinit(x^2 - 5);

The nfinit command initializes a structure containing a lot of information about the number field.

To find the ring of integers, we ask PARI/GP for an integral basis. The command is nfbasis. The output is a basis for $O_K$ as a $\mathbb{Z}$-module, expressed in terms of powers of the generator x (which represents $\sqrt{5}$ in this case).

nfbasis(x^2 - 5)

The output should be [1, 1/2*x + 1/2]. This corresponds to the basis $\{1,\frac{1+\sqrt{5}}{2}\}$, confirming that $O_{\mathbb{Q}(\sqrt{5})}=\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$.

Now try it for $K=\mathbb{Q}(\sqrt{3})$, where $d=3\equiv 3(\mathrm{mod}4)$. What do you expect the basis to be?

nfbasis(x^2 - 3)

The output here should be [1, x], corresponding to the basis $\{1,\sqrt{3}\}$, confirming that $O_{\mathbb{Q}(\sqrt{3})}=\mathbb{Z}[\sqrt{3}]$.

Feel free to experiment with other quadratic fields. This immediate computational feedback is one of the strengths of using a system like PARI/GP.

Conclusion

In this lesson, we have laid the essential groundwork for our study of class numbers. We've established our main objects of interest and their formal definitions:

• A number field $K$ is a finite field extension of $\mathbb{Q}$.
• An algebraic integer is a root of a monic polynomial with integer coefficients.
• The ring of integers $O_K$ of a number field $K$ is the set of all algebraic integers within $K$. It serves as the correct generalization of $\mathbb{Z}$ for doing number theory in $K$.

We saw that for quadratic fields $\mathbb{Q}(\sqrt{d})$, the ring of integers has a simple, explicit description that depends on $d(\mathrm{mod}4)$, and we learned how to verify this using PARI/GP.

In the next lesson, we will define the norm and trace of elements in a number field. These are two fundamental maps from $K$ down to $\mathbb{Q}$ that provide invaluable information about the arithmetic structure of the elements in $O_K$. You have already seen them appear in the proof for determining the ring of integers of a quadratic field, and we will now study them in their own right.