Representing Algebraic Numbers

Executive summary

Choose the representation that makes the dominant operation cheap

An element of a number field can be held as a polynomial in θ, as the matrix of multiplication by that element, as the vector of its images under all embeddings, or as its minimal polynomial. Multiplication is cheapest in the conjugate representation, trace and norm fall out of the matrix representation, and exactness lives in the standard representation. Practical systems keep more than one and convert as needed.

Learning objectives

Describe each of the four representations and its natural operations.
Convert between representations and state the cost of each conversion.
Choose a representation for a given computational profile.
Explain why the minimal polynomial is not a complete representation.

Section 01The four representations

Representations of an element of a number field of degree n
Representation	Data	Cheap operations	Expensive operations
Standard	Rational vector of coefficients against 1, θ, …, θⁿ⁻¹	Addition, scalar multiplication, exactness	Multiplication needs reduction modulo T; inversion needs extended GCD
Matrix (regular)	n×n rational matrix of multiplication by α	Trace, norm, characteristic polynomial, inversion	n² storage; multiplication is matrix multiplication
Conjugate vector	The n complex values σ_i(α)	Multiplication and division are componentwise	Not exact; addition is fine but recovery of exact coefficients needs precision
Minimal polynomial	The monic irreducible polynomial of α	Degree, conjugates, algebraic identity	Does not identify which root — incomplete on its own

The minimal polynomial is not enough

Two distinct elements of the same field — for instance √2 and −√2 — share a minimal polynomial. It identifies an element only up to conjugacy, so it must be paired with a root selection (a numerical approximation, or an expression in θ) to specify the element.

Section 02The regular representation

Multiplication by α is a ℚ-linear map on K. Its matrix M_α against a chosen basis is the regular representation, and it converts field questions into linear algebra.

Tr(α) = tr(M_α), N(α) = det(M_α), χ_α(x) = det(xI − M_α)

The map α ↦ M_α is a ring homomorphism, so products and inverses correspond to matrix products and inverses. Inversion in particular is just a linear solve, which is often more convenient than the extended Euclidean algorithm on polynomials.

Characteristic polynomial versus minimal polynomial

The characteristic polynomial of M_α is the minimal polynomial raised to the power [K : ℚ(α)]. They coincide exactly when α generates the whole field — which is the test for whether α is a primitive element.

Section 03Conversions

AlgorithmStandard to matrix representationin: α in standard form → out: the n×n regular matrix

For j = 0, …, n−1, compute α · θ^j as a polynomial in θ.
Reduce each product modulo the defining polynomial T. This is the only step that needs polynomial arithmetic.
Write each reduced product as a coefficient vector; these vectors are the columns of M_α.
Return M_α.

Conversion costs
From	To	Cost	Method
Standard	Matrix	O(n²)	Multiply by each basis element and reduce
Matrix	Standard	O(n)	Read the first column
Standard	Conjugate	O(n²) evaluations	Evaluate at each root
Conjugate	Standard	O(n²) plus precision	Interpolate, then round — requires certified precision
Standard	Minimal polynomial	O(n³)	Characteristic polynomial of the matrix, then remove repeated factors

Section 04Choosing a representation

What dominates the computation?
- Ring arithmetic on integers Integral basis coordinates — integer vectors, exact, and directly compatible with Hermite normal form.
- Many multiplications Conjugate vectors — componentwise, but only with a certified precision budget and an exact recovery step.
- Traces and norms Matrix representation — both are single linear algebra operations.
- Field membership and identity Standard representation — exact comparison of rational vectors.

Keep the exact form authoritative

Where a numerical representation is used for speed, the exact standard or integral-basis form should remain the source of truth, with the numerical form treated as a cache. Any result derived numerically must be confirmed exactly before it is recorded.

ReferenceFrequently asked questions

Why not always use the matrix representation?

Because it costs n² storage per element rather than n, and multiplication becomes matrix multiplication. It is the right choice when traces, norms or inverses dominate, and the wrong one when many elements must simply be stored and added.

How much precision do conjugate vectors need?

Enough that the exact coefficients can be recovered by rounding after interpolation. That depends on the size of the coefficients and on the conditioning of the root set, so the precision must be derived from an explicit bound rather than fixed in advance.

Can an element be represented against the integral basis instead of powers of theta?

Yes, and for algebraic integers it should be — the coordinates are then integers rather than rationals with a common denominator. All ideal and module arithmetic uses this representation.

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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 Representing Algebraic Numbers. 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 Representing Algebraic Numbers as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—representation, number, section, algebraic, matrix—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 Representing Algebraic Numbers?