Decomposition of Prime Numbers in Number Fields

Q: How do I know in advance whether p ramifies?

p ramifies exactly when it divides the field discriminant dK. Since only finitely many primes divide it, all but finitely many primes are unramified and fall under the simple algorithm.

Q: What is the residue field used for?

It is the finite field ℤK/𝔽, of size p^f. Reduction into it converts questions about algebraic integers into finite field computations — the basis of point counting on curves and of modular methods over number fields.

Executive summary

Factor the defining polynomial modulo p — usually that is the whole answer

A rational prime generates an ideal that factors into primes of ℤ_K with exponents e_i and residue degrees f_i satisfying ∑e_if_i = n. When p does not divide the index, Dedekind's theorem says the decomposition is read directly off the factorisation of the defining polynomial modulo p. The hard case is the finitely many primes dividing the index, which need the Buchmann–Lenstra machinery.

Learning objectives

State the fundamental identity and interpret e and f.
Apply Dedekind's theorem to decompose a prime.
Identify split, inert and ramified primes.
Compute the valuation of an element or ideal at a prime.
Recognise when the simple algorithm does not apply.

Section 01The fundamental identity

pℤ_K = ∏_i=1^g 𝔽_i^e_i, ∑_i e_i f_i = n

Here e_i is the ramification index and f_i the residue degree, the degree of ℤ_K/𝔽_i over F_p. The norm of 𝔽_i is p^f_i.

Decomposition types in a degree-n field
Type	Pattern	Meaning
Totally split	g = n, all e = f = 1	n distinct primes of norm p
Inert	g = 1, e = 1, f = n	p remains prime; one prime of norm pⁿ
Totally ramified	g = 1, e = n, f = 1	pℤ_K = 𝔽ⁿ
Partially split	1 < g < n	The general unramified case
Ramified	some e_i > 1	Occurs only for p dividing d_K — finitely many primes

Galois extensions are uniform

In a Galois extension the Galois group permutes the primes above p transitively, so all e_i are equal and all f_i are equal: efg = n. This rigidity is what makes decomposition in cyclotomic and other Galois fields predictable from congruence conditions alone.

Section 02Dedekind's theorem and the simple algorithm

If p does not divide the index [ℤ_K : ℤ[θ]], the decomposition mirrors the factorisation of the defining polynomial:

AlgorithmPrime decomposition for p not dividing the indexin: T, prime p not dividing the index → out: the primes above p

Factor T modulo p as ∏ g̅_i(x)^e_i with each g̅_i irreducible.
For each i, lift g̅_i arbitrarily to g_i ∈ ℤ[x].
Set 𝔽_i ← (p, g_i(θ)) — a two-element representation. The lift does not matter; different lifts give the same ideal.
Then e_i is the multiplicity and f_i = deg g̅_i.
Verify ∑ e_if_i = n as a consistency check.

Check the index condition

Applying this to a prime dividing the index produces ideals that are not prime, or misses primes entirely, with no internal signal of failure. Since only primes whose square divides disc(T) can divide the index, the check is cheap — but it must be made.

Section 03Valuations

The valuation v_𝔽(α) is the exponent of 𝔽 in the factorisation of (α). It is computed without factoring the norm by using an element π that is a uniformiser at 𝔽 and a unit at the other primes above p.

AlgorithmValuation at a prime idealin: α ∈ ℤ_K, prime 𝔽 → out: v_𝔽(α)

Precompute π ∈ 𝔽 \ 𝔽² with π ∉ 𝔽_j for j ≠ i. Constructed once per prime by the Chinese remainder theorem.
Set v ← 0 and β ← α.
While β ∈ 𝔽: set β ← βπ⁻¹ · p (kept integral), and increment v.
Return v.

Where valuations are needed

Relation collection in class group algorithms produces elements whose ideals must be expressed as exponent vectors over the factor base. Each entry of such a vector is a valuation, so this routine runs enormously often and is worth optimising.

Section 04The hard primes

For p dividing the index, Dedekind's theorem does not apply. Two approaches exist, and both are covered in the Number Fields II stream.

Approach

Compute the maximal order first

Run the Round 2 algorithm to obtain ℤ_K, then decompose using the structure of ℤ_K/pℤ_K directly.

Approach

Buchmann–Lenstra

Decompose the separable algebra ℤ_K/pℤ_K into local factors without first computing a full integral basis.

Approach

Newton polygons

Read ramification data off the Newton polygon of T at p. Often gives the answer immediately when the polygon has distinct slopes.

ReferenceFrequently asked questions

How do I know in advance whether p ramifies?

p ramifies exactly when it divides the field discriminant d_K. Since only finitely many primes divide it, all but finitely many primes are unramified and fall under the simple algorithm.

Do the exponents in the fundamental identity depend on the defining polynomial?

No. The e and f are invariants of the field and the prime. Different defining polynomials give different intermediate factorisations but the same decomposition type, which makes the identity a useful cross-check between representations.

What is the residue field used for?

It is the finite field ℤ_K/𝔽, of size p^f. Reduction into it converts questions about algebraic integers into finite field computations — the basis of point counting on curves and of modular methods over number fields.

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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 Decomposition of Prime Numbers in 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 Decomposition of Prime Numbers in Number Fields as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—section, decomposition, prime, number, fields—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 Decomposition of Prime Numbers in Number Fields?