Number Fields and Algebraic Numbers
The Conjugate Vector Representation
Representing a field element by its images under all embeddings, the analytic information this exposes, and the precision it demands.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0577
The conjugate vector lists the images of an element under every embedding into the complex numbers. It is numerical rather than exact, and it exposes exactly the information the exact representations hide: size, sign, and analytic behaviour.
The representation
What it exposes
- Norm and trace
- The product and sum of the entries. A useful cross-check against the exact values from the matrix representation.
- Size
- The largest absolute value among the entries — the natural measure of how big an algebraic number is, invisible in the standard representation.
- Unit detection
- An algebraic integer is a unit exactly when the product of the absolute values is one.
- Logarithmic embedding
- The logarithms of the absolute values, weighted by embedding type. See the logarithmic embedding.
Precision
| Use | Precision requirement |
|---|---|
| Comparing sizes | Modest |
| Regulator computation | High; the determinant is of logarithms |
| Recovering an exact element | Very high; see dependence detection |
| Deciding equality or vanishing | Not valid at any precision; use an exact representation |
Obtaining the vector
Compute the complex roots of the defining polynomial once, to the required precision, then evaluate the element's polynomial at each root. The root computation is the expensive part and is amortised across all elements of the field.
Where it is essential
The conjugate vector is required wherever archimedean information matters: computing regulators, performing ideal reduction against a size measure, and evaluating analytic class number formulas — see analytic formulas.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.2.4. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.