Norm and Trace in Number Fields

A Linear Algebra Perspective

Since a number field $K$ of degree $n=[K:\mathbb{Q}]$ is an $n$-dimensional $\mathbb{Q}$-vector space, any element $\alpha \in K$ gives rise to a $\mathbb{Q}$-linear map. This map, often denoted $m_{\alpha }$, is simply multiplication by $\alpha$:

Because this is a linear transformation on a finite-dimensional vector space, it can be represented by an $n\times n$ matrix once we choose a basis for $K$ over $\mathbb{Q}$. The trace and determinant of this matrix are invariants of the linear transformation itself and do not depend on the choice of basis. This gives us our first definition.

The following video provides a concise introduction to this concept.

With this, we can state the formal definition:

Definition: Let $K$ be a number field and $\alpha \in K$. The trace of $\alpha$, denoted ${\text{Tr}}_{K/\mathbb{Q}}(\alpha )$, is the trace of the matrix of the linear map $m_{\alpha }$. The norm of $\alpha$, denoted $N_{K/\mathbb{Q}}(\alpha )$, is the determinant of the matrix of $m_{\alpha }$.

To make this concrete, let's work through the calculations. The following reading from Keith Conrad's notes provides detailed examples.

As you saw in the reading, for $K=\mathbb{Q}(\sqrt{d})$ and $\alpha =a+b\sqrt{d}$, the matrix of $m_{\alpha }$ with respect to the basis $\{1,\sqrt{d}\}$ is

From this, we immediately get:

• ${\text{Tr}}_{K/\mathbb{Q}}(a+b\sqrt{d})=a+a=2a$
• $N_{K/\mathbb{Q}}(a+b\sqrt{d})=a^2-b^{2}d$

An important fact, which we will use frequently, is that if $\alpha$ is an algebraic integer in $O_K$, then its norm and trace are always rational integers (${\text{Tr}}_{K/\mathbb{Q}}(\alpha )\in \mathbb{Z}$ and $N_{K/\mathbb{Q}}(\alpha )\in \mathbb{Z}$).

Properties and Alternative Formulations

The linear algebra definition directly implies two fundamental properties, which follow from the identities $\text{Tr}(A+B)=\text{Tr}(A)+\text{Tr}(B)$ and $\det (AB)=\det (A)\det (B)$.

• The trace is $\mathbb{Q}$-linear:
  • ${\text{Tr}}_{K/\mathbb{Q}}(\alpha +\beta )={\text{Tr}}_{K/\mathbb{Q}}(\alpha )+{\text{Tr}}_{K/\mathbb{Q}}(\beta )$
  • ${\text{Tr}}_{K/\mathbb{Q}}(c\alpha )=c\cdot {\text{Tr}}_{K/\mathbb{Q}}(\alpha )$ for any $c\in \mathbb{Q}$
• The norm is multiplicative:
  • $N_{K/\mathbb{Q}}(\alpha \beta )=N_{K/\mathbb{Q}}(\alpha )N_{K/\mathbb{Q}}(\beta )$

These properties are extremely useful, but calculating the full matrix for $m_{\alpha }$ can be tedious for high-degree fields. Fortunately, there are more direct connections to polynomials and field embeddings.

Connection to Polynomials

The norm and trace of an element $\alpha$ are encoded in its characteristic polynomial, ${\chi }_{\alpha ,K/\mathbb{Q}}(X)=\det (X\cdot I-[m_{\alpha }])$. Specifically,

This characteristic polynomial is, in turn, directly related to the minimal polynomial of $\alpha$ over $\mathbb{Q}$, which you know from basic field theory. The relationship is simple and powerful: if ${\mu }_{\alpha ,\mathbb{Q}}(X)$ is the minimal polynomial of $\alpha$ and has degree $d=[\mathbb{Q}(\alpha ):\mathbb{Q}]$, then

The following reading explains this relationship and derives formulas for the norm and trace directly from the coefficients of the minimal polynomial.

Connection to Embeddings

Perhaps the most elegant and often most practical way to compute the norm and trace is via the embeddings of the number field into the complex numbers. For a number field $K$ of degree $n$, there are exactly $n$ distinct embeddings ${\sigma }_i:K\hookrightarrow \mathbb{C}$. The collection of these embeddings reveals the "conjugates" of any element in $K$.

Definition: For $\alpha \in K$, its norm and trace are the product and sum of its images under all embeddings of $K$ into $\mathbb{C}$:

Let's see why this elegant formulation works.

Let's revisit our quadratic field example, $K=\mathbb{Q}(\sqrt{d})$, where $d$ is a square-free integer. The two embeddings are the identity map, ${\sigma }_1(a+b\sqrt{d})=a+b\sqrt{d}$, and the conjugation map, ${\sigma }_2(a+b\sqrt{d})=a-b\sqrt{d}$.
Using the embedding formulas:

• ${\text{Tr}}_{K/\mathbb{Q}}(a+b\sqrt{d})={\sigma }_1(a+b\sqrt{d})+{\sigma }_2(a+b\sqrt{d})=(a+b\sqrt{d})+(a-b\sqrt{d})=2a$
• $N_{K/\mathbb{Q}}(a+b\sqrt{d})={\sigma }_1(a+b\sqrt{d})\cdot {\sigma }_2(a+b\sqrt{d})=(a+b\sqrt{d})(a-b\sqrt{d})=a^2-b^{2}d$

This matches our linear algebra calculation perfectly but was significantly faster. This formulation is often the most direct path to computation.

Computation in PARI/GP

Now, let's use PARI/GP to verify these calculations. The relevant commands are nfelttrace and nfeltnorm. They take a number field structure (from nfinit) and a field element as arguments.

Let's test this on a cubic field, say $K=\mathbb{Q}(\gamma )$ where ${\gamma }^3-\gamma -1=0$. We will compute the norm and trace of the element $\beta ={\gamma }^2$.
From Conrad's Example 3.3, we find the matrix for $m_{{\gamma }^2}$ is:

From this matrix, we can see:

• ${\text{Tr}}_{K/\mathbb{Q}}({\gamma }^2)=0+1+1=2$
• $N_{K/\mathbb{Q}}({\gamma }^2)=\det ([m_{{\gamma }^2}])=0-1(0-1)+0=1$

Now, let's verify this in PARI/GP.

1. Initialize the field: K = nfinit(x^3 - x - 1);
2. Define the element: beta = x^2;
3. Compute the trace: nfelttrace(K, beta)
4. Compute the norm: nfeltnorm(K, beta)

You should see 2 and 1 as the outputs, respectively. Feel free to try another element, like gamma + 2, and compute its trace and norm by hand using linearity, then check your result in PARI/GP.
$\text{Tr}(\gamma +2)=\text{Tr}(\gamma )+\text{Tr}(2)$. From the minimal polynomial $x^3-x-1$, $\text{Tr}(\gamma )=0$. The trace of a rational number $c$ is $n\cdot c$, so $\text{Tr}(2)=3\cdot 2=6$. Thus, $\text{Tr}(\gamma +2)=6$.

Conclusion

In this lesson, we have defined and learned to compute the norm and trace of elements in a number field. We've explored three interconnected perspectives:

1. Linear Algebra: The trace and determinant of the multiplication-by-$\alpha$ map.
2. Polynomials: The coefficients of the characteristic polynomial, which is a power of the minimal polynomial.
3. Embeddings: The sum and product of the Galois conjugates of an element.

These tools are indispensable for investigating the arithmetic of number fields. The norm, in particular, will be a constant companion as we begin to study ideals. It allows us to measure the "size" of an algebraic integer, reducing questions about divisibility in $O_K$ to questions about divisibility in $\mathbb{Z}$.

In our next lesson, we will use the trace to define another fundamental invariant of a number field: the discriminant. The discriminant is a single number that packs a surprising amount of information about the ring of integers $O_K$, including a first glimpse into how prime numbers from $\mathbb{Z}$ behave when they are considered in this larger ring.