Descent to ℚ, and what survives the descent
The trace is the sum of the conjugates and the norm their product. Both land in ℚ, and in ℤ when the element is an algebraic integer. The norm is multiplicative, which makes it the primary tool for detecting units and for defining the norm of an ideal; the trace is additive and defines the trace form whose determinant is the discriminant. Both are computable exactly from the regular representation, with no reference to complex embeddings.
Learning objectives
- Define trace and norm in terms of embeddings and of the regular representation.
- Use multiplicativity of the norm to detect units.
- Relate the coefficients of the characteristic polynomial to trace and norm.
- Apply the trace form to compute a discriminant.
- Use norms as a cheap consistency check on ideal arithmetic.
Section 01Definitions
Equivalently, and exactly, they are the trace and determinant of the multiplication matrix Mα. That second definition is the one to implement: it is exact, requires no complex arithmetic, and generalises without change to relative extensions.
| Property | Trace | Norm |
|---|---|---|
| Behaviour | Additive: Tr(α+β) = Tr(α) + Tr(β) | Multiplicative: N(αβ) = N(α)N(β) |
| On ℚ | Tr(a) = na | N(a) = an |
| On algebraic integers | Lands in ℤ | Lands in ℤ |
| Relation to χ | −(coefficient of xn−1) | (−1)n × constant term |
| Degeneracy | Trace form is non-degenerate in characteristic 0 | N(α) = 0 only for α = 0 |
Section 02Detecting units
An algebraic integer is a unit of ℤK exactly when its norm is ±1. The proof is immediate from multiplicativity: if αβ = 1 then N(α)N(β) = 1 with both factors in ℤ.
One determinant decides unit status. This test underlies the whole of unit group computation: candidate units are produced by other means — continued fractions, relation collection, ideal reduction — and this is the filter that confirms them.
The same reasoning gives the norm of an ideal: N(𝔼) = [ℤK : 𝔼], which for a principal ideal (α) equals |N(α)|. Multiplicativity persists, so norms provide a running consistency check on ideal arithmetic at negligible cost.
- Compute the product ideal 𝔼𝔽 by multiplying generators and taking the Hermite normal form.
- Read N(𝔼𝔽) from the product of the HNF diagonal entries.
- Compare against N(𝔼) · N(𝔽). Any mismatch indicates an error in the multiplication or the HNF.
- Proceed only if they agree.
Section 03The trace form and the discriminant
The trace defines a symmetric bilinear form on K as a ℚ-vector space:
Its Gram matrix against a basis has determinant equal to the discriminant of that basis. For an integral basis this is the field discriminant dK, and the change of basis behaviour explains why the discriminant of a suborder differs by a perfect square:
Computing disc(ℤ[θ]) is easy — it is the polynomial discriminant. Extracting dK requires identifying the square factors, which is why determining the ring of integers is as hard as factoring the discriminant, and why the Round 2 algorithm is organised prime by prime.
ReferenceFrequently asked questions
Can trace and norm be computed from numerical conjugates?
They can, and the result is a useful cross-check, but the authoritative computation should be the exact one via the regular representation. The numerical value must round to an integer for an algebraic integer, and a failure to do so is a precision warning rather than a result.
What is the trace of a root of unity?
For a primitive n-th root of unity in the cyclotomic field, the trace is the Moebius function μ(n). It is a classical identity and a convenient test case for an implementation.
Does the norm detect irreducibility of an element?
A prime norm implies the element generates a prime ideal and is therefore irreducible. The converse fails: an irreducible element can have composite norm, and in a field with class number greater than 1 irreducible elements need not generate prime ideals at all.
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