Norm and Trace

Two Fundamental Invariants

Let $K$ be a number field of degree

$$ [K:\mathbb{Q}]=n. $$

Each element $\alpha\in K$ has two fundamental rational invariants: its norm and its trace. These generalize familiar operations from quadratic fields.

The norm behaves like a product of conjugates. The trace behaves like a sum of conjugates.

If the embeddings of $K$ into $\mathbb{C}$ are

$$ \sigma_1,\sigma_2,\ldots,\sigma_n, $$

then the norm and trace are defined by

$$ N_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)\sigma_2(\alpha)\cdots\sigma_n(\alpha), $$

and

$$ \operatorname{Tr}_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)+\sigma_2(\alpha)+\cdots+\sigma_n(\alpha). $$

These quantities belong to $\mathbb{Q}$. If $\alpha$ is an algebraic integer, then both belong to $\mathbb{Z}$.

Quadratic Fields

Let

$$ K=\mathbb{Q}(\sqrt d), $$

where $d$ is squarefree. Every element has the form

$$ \alpha=a+b\sqrt d, \qquad a,b\in\mathbb{Q}. $$

There are two embeddings into $\mathbb{C}$:

$$ \sqrt d\mapsto \sqrt d, \qquad \sqrt d\mapsto -\sqrt d. $$

Hence the conjugate of $\alpha$ is

$$ \overline{\alpha}=a-b\sqrt d. $$

The norm is

$$ N(\alpha) = \alpha\overline{\alpha} = (a+b\sqrt d)(a-b\sqrt d) = a^2-db^2. $$

The trace is

$$ \operatorname{Tr}(\alpha) = \alpha+\overline{\alpha} = 2a. $$

These formulas are the simplest model for the general theory.

Example: Gaussian Integers

In the Gaussian field

$$ \mathbb{Q}(i), $$

an element has the form

$$ \alpha=a+bi. $$

Its conjugate is

$$ \overline{\alpha}=a-bi. $$

Therefore

$$ N(a+bi)=a^2+b^2, $$

and

$$ \operatorname{Tr}(a+bi)=2a. $$

The norm explains why sums of two squares appear naturally in Gaussian integer arithmetic.

For example,

$$ N(3+4i)=3^2+4^2=25. $$

Since the norm is multiplicative,

$$ N(\alpha\beta)=N(\alpha)N(\beta), $$

the product of two sums of squares is again a sum of squares.

Norm as Determinant

There is another useful definition of the norm.

Multiplication by $\alpha$ defines a $\mathbb{Q}$-linear map

$$ m_\alpha:K\to K, \qquad x\mapsto \alpha x. $$

Since $K$ is an $n$-dimensional vector space over $\mathbb{Q}$, this map has a determinant.

One has

$$ N_{K/\mathbb{Q}}(\alpha)=\det(m_\alpha). $$

Similarly,

$$ \operatorname{Tr}{K/\mathbb{Q}}(\alpha)=\operatorname{trace}(m\alpha). $$

This linear algebra viewpoint is often the cleanest definition in abstract settings.

Multiplicativity and Additivity

The norm is multiplicative:

$$ N(\alpha\beta)=N(\alpha)N(\beta). $$

This follows either from the embedding definition or from the determinant interpretation.

The trace is additive:

$$ \operatorname{Tr}(\alpha+\beta) = \operatorname{Tr}(\alpha)+\operatorname{Tr}(\beta). $$

It is also compatible with rational scaling:

$$ \operatorname{Tr}(q\alpha)=q\operatorname{Tr}(\alpha) $$

for $q\in\mathbb{Q}$.

Thus the norm behaves like a multiplicative size, while the trace behaves like a linear measurement.

Minimal Polynomials

Suppose $\alpha$ has minimal polynomial

$$ m_\alpha(x) = x^r+c_{r-1}x^{r-1}+\cdots+c_1x+c_0. $$

If $K=\mathbb{Q}(\alpha)$, then the conjugates of $\alpha$ are the roots of this polynomial.

Thus

$$ \operatorname{Tr}{K/\mathbb{Q}}(\alpha)=-c{r-1}, $$

and

$$ N_{K/\mathbb{Q}}(\alpha)=(-1)^r c_0. $$

For example, $\sqrt2$ has minimal polynomial

$$ x^2-2. $$

The trace is

$$ 0, $$

and the norm is

$$ -2. $$

Indeed,

$$ N(\sqrt2)=(\sqrt2)(-\sqrt2)=-2. $$

Algebraic Integers

If

$$ \alpha\in\mathcal O_K, $$

then all conjugates of $\alpha$ are algebraic integers. Their sum and product are rational algebraic integers, hence ordinary integers.

Therefore

$$ N_{K/\mathbb{Q}}(\alpha)\in\mathbb{Z}, \qquad \operatorname{Tr}_{K/\mathbb{Q}}(\alpha)\in\mathbb{Z}. $$

This makes norm and trace useful arithmetic tools.

For example, if $\alpha\in\mathcal O_K$ and

$$ N(\alpha)=\pm1, $$

then $\alpha$ is a unit.

Norms and Divisibility

Norms often reduce divisibility questions in $\mathcal O_K$ to divisibility questions in $\mathbb{Z}$.

If

$$ \alpha\mid\beta $$

in $\mathcal O_K$, then

$$ N(\alpha)\mid N(\beta) $$

in $\mathbb{Z}$, up to sign.

This gives a useful obstruction. For example, in the Gaussian integers, if

$$ \alpha\mid 5, $$

then

$$ N(\alpha)\mid N(5)=25. $$

Thus possible norms of divisors are restricted to divisors of $25$.

Trace Pairing

The trace defines a bilinear form on $K$:

$$ (\alpha,\beta)\mapsto \operatorname{Tr}_{K/\mathbb{Q}}(\alpha\beta). $$

This form is fundamental in the study of discriminants and integral bases.

Given a basis

$$ \omega_1,\ldots,\omega_n $$

of $K$ over $\mathbb{Q}$, one forms the matrix

$$ \left(\operatorname{Tr}(\omega_i\omega_j)\right)_{i,j}. $$

Its determinant is the discriminant of the basis.

Thus trace connects field arithmetic with lattice geometry.

Geometric Meaning

Under the embeddings

$$ \sigma_i:K\to\mathbb{C}, $$

an element $\alpha$ becomes a tuple of complex numbers:

$$ (\sigma_1(\alpha),\ldots,\sigma_n(\alpha)). $$

The norm is their product. The trace is their sum.

When restricted to the ring of integers, these embeddings place $\mathcal O_K$ as a lattice in Euclidean space. Norm and trace then measure arithmetic information through this geometric embedding.

Arithmetic Role

Norm and trace are among the most important tools in algebraic number theory.

They are used to study:

• units,
• divisibility,
• ideals,
• discriminants,
• integral bases,
• ramification,
• field extensions.

The norm converts multiplication in a number field into multiplication in $\mathbb{Q}$. The trace converts addition and linear structure in a number field into ordinary rational data. Together, they provide the basic bridge between arithmetic inside $K$ and arithmetic over $\mathbb{Q}$.