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.
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.
Closed under addition and multiplication but not inversion. The algebraic integers inside a number field K form the ring of integers ℤK.
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 ℤK 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).
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 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.
| 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 2r1 (2π)r2 |
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.
Real quadratic
Signature (2, 0). Unit rank 1 — a single fundamental unit, whose logarithm is the regulator, computed classically by continued fractions.
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 ℤK, 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.
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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Curated next steps from this page. The site also surfaces algorithmically related reading below.
- Number Fields IRepresenting Algebraic Numbers
- Number Fields IDiscriminants and Integral Bases
- Number Fields ITrace, Norm and the Characteristic Polynomial
- Polynomial AlgorithmsRoot Finding over the Complex Numbers
- Number Fields IIComputing Galois Groups of Number Fields
- Number Fields IOrders and Ideals in Number Fields
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.