Algebraic Numbers and Number Fields

Executive summary

A number field is a defining polynomial plus everything that follows from it

A number field is a finite extension of ℚ, and by the primitive element theorem it is always ℚ(θ) for a single algebraic number θ. Computationally the field is the minimal polynomial of θ, and every subsequent computation — integral basis, ideals, class group, units — is derived from it. The degree, the signature and the discriminant are the three coarse invariants that shape everything that follows.

Learning objectives

Define algebraic numbers and algebraic integers and distinguish them.
State the primitive element theorem and its computational significance.
Compute the real and complex embeddings of a number field.
Explain how the signature determines the rank of the unit group.
Recognise that a field has many defining polynomials and why the choice matters.

Section 01Algebraic numbers and integers

An algebraic number is a root of a non-zero polynomial with rational coefficients; an algebraic integer is a root of a monic polynomial with integer coefficients. The minimal polynomial is the monic generator of the ideal of polynomials vanishing at α, and it is irreducible.

Fact 1Algebraic numbers form a field

Sums, products and inverses of algebraic numbers are algebraic. The degree of the result is bounded by the product of the degrees, and is computed via resultants.

Fact 2Algebraic integers form a ring

Closed under addition and multiplication but not inversion. The algebraic integers inside a number field K form the ring of integers ℤ_K.

The rational test

An algebraic integer that is rational is an ordinary integer. This apparently modest fact is used constantly: it is why norms and traces of algebraic integers are in ℤ, and why an element of ℤ_K with rational norm ±1 is a unit.

Section 02Number fields and the primitive element theorem

A number field K is a field containing ℚ that is finite-dimensional as a ℚ-vector space. That dimension is the degree n = [K : ℚ]. Since characteristic zero fields are separable, the primitive element theorem applies: there is a single θ with K = ℚ(θ).

K ≅ ℚ[x] / (T(x)), T irreducible of degree n

Every element is then a polynomial in θ of degree below n with rational coefficients — a vector of n rationals. This is the fundamental data structure of the entire subject.

The defining polynomial is not canonical

The same field has infinitely many defining polynomials, and they differ enormously in usefulness. A polynomial with large coefficients makes every subsequent computation more expensive and can make the integral basis computation intractable. Reducing the defining polynomial — finding a small equivalent one by LLL on the ring of integers — is a standard preprocessing step, not an optional polish.

Section 03Embeddings and the signature

A number field of degree n has exactly n embeddings into ℂ, one for each complex root of the defining polynomial. Those landing in ℝ are the real embeddings; the remainder come in conjugate pairs.

n = r₁ + 2r₂, signature (r₁, r₂)

What the signature controls
Quantity	Depends on signature how
Unit rank	r₁ + r₂ − 1, by Dirichlet's theorem
Sign of the discriminant	Negative exactly when r₂ is odd
Roots of unity	Only ±1 when r₁ > 0
Regulator	Defined as a determinant of size r₁ + r₂ − 1
Minkowski bound	Includes the factor (4/π)^r₂
Class number formula	Residue involves 2^r₁ (2π)^r₂

Example

Imaginary quadratic

Signature (0, 1). Unit rank 0 — the unit group is finite, which is why these fields are the easy case for class group computation.

Example

Real quadratic

Signature (2, 0). Unit rank 1 — a single fundamental unit, whose logarithm is the regulator, computed classically by continued fractions.

Example

Cyclotomic field of p-th roots

Signature (0, (p−1)/2) for odd prime p. Totally complex, with unit rank (p−3)/2.

Section 04Computational representation

Stage 01Defining polynomialA monic irreducible T ∈ ℤ[x], preferably with small coefficients.
Stage 02Root approximationsAll n complex roots to certified precision, giving the embeddings.
Stage 03Integral basisA ℤ-basis of ℤ_K, expressed as rational combinations of powers of θ with a common denominator.
Stage 04Multiplication tableProducts of basis elements in terms of the basis — enabling all ring arithmetic as matrix operations.

Two coordinate systems, always

Elements are held either as polynomials in θ over ℚ, or as integer vectors against the integral basis. The first is natural for field arithmetic, the second for ideal and module work. Confusing them is the single most common bug in number field code, so the coordinate system should be part of the type, not a convention.

ReferenceFrequently asked questions

Is every number field a subfield of the complex numbers?

Abstractly a number field is a field extension, but it has n distinct embeddings into ℂ and no canonical one. Fixing an embedding is a choice, and it matters: statements about positivity or about the size of an element are meaningful only relative to a chosen embedding.

Why reduce the defining polynomial?

Because coefficient size propagates. A defining polynomial with hundred-digit coefficients produces an integral basis with hundred-digit denominators, ideals with huge Hermite normal forms, and a class group computation that may not finish. Reduction is cheap relative to what it saves.

How do I tell whether two defining polynomials give the same field?

Test whether each has a root in the field defined by the other, by factoring one polynomial over the other's field. If both directions succeed the fields are isomorphic, and the factorisation supplies the isomorphism explicitly.

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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Algebraic Numbers and Number Fields. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Algebraic Numbers and Number Fields as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—number, algebraic, fields, section, numbers—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Algebraic Numbers and Number Fields?