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Discriminants

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.

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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)=an∏i=1n(x−αi) f(x)=a_n\prod_{i=1}^n (x-\alpha_i)

be a polynomial with roots

α1,…,αn. \alpha_1,\dots,\alpha_n.

The discriminant of ff is

Δ(f)=an2n−2∏i<j(αi−αj)2. \Delta(f) = a_n^{2n-2} \prod_{i<j}(\alpha_i-\alpha_j)^2.

Δ(f)=an2n−2∏i<j(αi−αj)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

x2−d x^2-d

has roots

±d. \pm\sqrt d.

Its discriminant is

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

Discriminant of a Number Field

Let KK be a number field of degree nn, and let

ω1,…,ωn \omega_1,\dots,\omega_n

be an integral basis of

OK. \mathcal O_K.

Let

σ1,…,σn \sigma_1,\dots,\sigma_n

be the embeddings of KK into C\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 KK, denoted

ΔK, \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=Q(d), K=\mathbb{Q}(\sqrt d),

where dd is squarefree.

Then:

  • if
d≡1(mod4), d\equiv1\pmod4,

the discriminant is

ΔK=d, \Delta_K=d,
  • otherwise,
ΔK=4d. \Delta_K=4d.

For example:

FieldDiscriminant
Q(i)\mathbb{Q}(i)−4-4
Q(2)\mathbb{Q}(\sqrt2)88
Q(5)\mathbb{Q}(\sqrt5)55
Q(−3)\mathbb{Q}(\sqrt{-3})−3-3

These small integers encode deep arithmetic information.

Example: Gaussian Integers

Consider

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

An integral basis is

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

The embeddings are:

σ1(i)=i,σ2(i)=−i. \sigma_1(i)=i, \qquad \sigma_2(i)=-i.

Thus the embedding matrix is

(1i1−i). \begin{pmatrix} 1 & i\\ 1 & -i \end{pmatrix}.

Its determinant is

−2i. -2i.

Squaring gives

(−2i)2=−4. (-2i)^2=-4.

Hence

ΔQ(i)=−4. \Delta_{\mathbb{Q}(i)}=-4.

Trace Form Interpretation

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

Define the bilinear form

(α,β)↦Tr⁡K/Q(αβ). (\alpha,\beta)\mapsto \operatorname{Tr}_{K/\mathbb{Q}}(\alpha\beta).

For an integral basis

ω1,…,ωn, \omega_1,\dots,\omega_n,

one forms the matrix

(Tr⁡(ωiωj)). \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 pp ramifies in KK precisely when

p∣ΔK. p\mid \Delta_K.

For example, in

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

the discriminant is

−4. -4.

Thus only the prime 22 ramifies.

Indeed,

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

inside

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

Ramification describes how prime factorization changes inside number fields.

Change of Basis

Suppose two bases are related by an integer matrix AA:

ωi′=∑jaijωj. \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 KK,

OK \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>0B>0, there exist only finitely many number fields with

∣ΔK∣<B. |\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.