# Discriminants

## Measuring Arithmetic Complexity

The discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic.

Roughly speaking, the discriminant detects:

- linear dependence among embeddings,
- ramification of primes,
- geometric density of the ring of integers,
- arithmetic complexity of the field.

It appears throughout algebraic number theory, arithmetic geometry, and analytic number theory.

## Discriminant of a Polynomial

Let

$$
f(x)=a_n\prod_{i=1}^n (x-\alpha_i)
$$

be a polynomial with roots

$$
\alpha_1,\dots,\alpha_n.
$$

The discriminant of $f$ is

$$
\Delta(f) =
a_n^{2n-2}
\prod_{i<j}(\alpha_i-\alpha_j)^2.
$$

$$
\Delta(f)=a_n^{2n-2}\prod_{i<j}(\alpha_i-\alpha_j)^2
$$

The discriminant vanishes precisely when two roots coincide.

Thus it measures how distinct the roots are.

For example, the polynomial

$$
x^2-d
$$

has roots

$$
\pm\sqrt d.
$$

Its discriminant is

$$
(2\sqrt d)^2=4d.
$$

## Discriminant of a Number Field

Let $K$ be a number field of degree $n$, and let

$$
\omega_1,\dots,\omega_n
$$

be an integral basis of

$$
\mathcal O_K.
$$

Let

$$
\sigma_1,\dots,\sigma_n
$$

be the embeddings of $K$ into $\mathbb{C}$.

The discriminant of the basis is

$$
\Delta(\omega_1,\dots,\omega_n)
=
\det(\sigma_i(\omega_j))^2.
$$

Equivalently,

$$
\Delta(\omega_1,\dots,\omega_n)
=
\det\left(
\operatorname{Tr}(\omega_i\omega_j)
\right).
$$

The discriminant of the field $K$, denoted

$$
\Delta_K,
$$

is the discriminant of any integral basis.

A remarkable theorem states that this value is independent of the chosen integral basis.

## Quadratic Fields

The simplest examples occur in quadratic fields.

Let

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

where $d$ is squarefree.

Then:

- if

$$
d\equiv1\pmod4,
$$

the discriminant is

$$
\Delta_K=d,
$$

- otherwise,

$$
\Delta_K=4d.
$$

For example:

| Field | Discriminant |
|---|---|
| $\mathbb{Q}(i)$ | $-4$ |
| $\mathbb{Q}(\sqrt2)$ | $8$ |
| $\mathbb{Q}(\sqrt5)$ | $5$ |
| $\mathbb{Q}(\sqrt{-3})$ | $-3$ |

These small integers encode deep arithmetic information.

## Example: Gaussian Integers

Consider

$$
K=\mathbb{Q}(i).
$$

An integral basis is

$$
\{1,i\}.
$$

The embeddings are:

$$
\sigma_1(i)=i,
\qquad
\sigma_2(i)=-i.
$$

Thus the embedding matrix is

$$
\begin{pmatrix}
1 & i\\
1 & -i
\end{pmatrix}.
$$

Its determinant is

$$
-2i.
$$

Squaring gives

$$
(-2i)^2=-4.
$$

Hence

$$
\Delta_{\mathbb{Q}(i)}=-4.
$$

## Trace Form Interpretation

The discriminant can also be viewed geometrically through the trace pairing.

Define the bilinear form

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

For an integral basis

$$
\omega_1,\dots,\omega_n,
$$

one forms the matrix

$$
\left(
\operatorname{Tr}(\omega_i\omega_j)
\right).
$$

The discriminant is the determinant of this matrix.

Thus the discriminant measures the volume of the lattice formed by the ring of integers inside Euclidean space.

## Ramification of Primes

One of the most important roles of the discriminant is detecting ramification.

A prime number $p$ ramifies in $K$ precisely when

$$
p\mid \Delta_K.
$$

For example, in

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

the discriminant is

$$
-4.
$$

Thus only the prime $2$ ramifies.

Indeed,

$$
(2)=(1+i)^2
$$

inside

$$
\mathbb{Z}[i].
$$

Ramification describes how prime factorization changes inside number fields.

## Change of Basis

Suppose two bases are related by an integer matrix $A$:

$$
\omega_i'=\sum_j a_{ij}\omega_j.
$$

Then the discriminants satisfy

$$
\Delta(\omega_1',\dots,\omega_n')
=
(\det A)^2
\Delta(\omega_1,\dots,\omega_n).
$$

Thus changing bases modifies the discriminant only by a square factor.

For integral bases, the discriminant becomes an invariant of the field itself.

## Minkowski Geometry

Under the embeddings of $K$,

$$
\mathcal O_K
$$

becomes a lattice in Euclidean space.

The discriminant controls the covolume of this lattice.

Fields with small discriminant correspond to dense arithmetic lattices.

This geometric interpretation is fundamental in Minkowski theory and geometry of numbers.

## Analytic Number Theory

The discriminant appears in analytic formulas involving Dedekind zeta functions.

The class number formula contains:

- the discriminant,
- the regulator,
- the class number,
- the number of roots of unity.

Thus the discriminant governs analytic growth and arithmetic density simultaneously.

## Bounds and Finiteness

A major theorem states:

**Theorem (Hermite).** For every integer $B>0$, there exist only finitely many number fields with

$$
|\Delta_K|<B.
$$

Thus the discriminant measures arithmetic complexity strongly enough to control finiteness.

This theorem is foundational in arithmetic classification problems.

## Structural Importance

The discriminant connects many major concepts:

- embeddings,
- trace pairings,
- ramification,
- geometry of lattices,
- analytic behavior of zeta functions,
- classification of number fields.

It is one of the central numerical invariants in algebraic number theory.

In many ways, the discriminant plays the role of an arithmetic curvature measuring how far a number field deviates from ordinary rational arithmetic.