Number Fields and Algebraic Numbers
Trace, Norm and the Characteristic Polynomial
Trace, norm and characteristic polynomial of a field element, their computation, and their use as invariants and cross-checks.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0578
Trace and norm are the two most useful scalar invariants of a field element. They are the extreme coefficients of the characteristic polynomial, and both are computable in several ways, which makes them excellent cross-checks.
Definitions
| Route | Trace | Norm |
|---|---|---|
| Matrix representation | Trace of the matrix | Determinant of the matrix |
| Conjugate vector | Sum of entries | Product of entries |
| Characteristic polynomial | Negative of the second coefficient | Constant term, up to sign |
| Resultant | Not directly | Resultant of the element's polynomial with the defining polynomial |
Properties
- Trace is additive
- The trace of a sum is the sum of traces; it is a linear map to the rationals.
- Norm is multiplicative
- The norm of a product is the product of norms. This is what makes the norm useful for factorisation arguments.
- Integrality
- For an algebraic integer, both are rational integers.
- Units
- An algebraic integer is a unit exactly when its norm is plus or minus one.
Characteristic versus minimal polynomial
The characteristic polynomial of the multiplication matrix always has degree n. It equals the minimal polynomial raised to the power n divided by the degree of the element.
The trace form
The bilinear form sending a pair of elements to the trace of their product is non-degenerate for a separable extension. Its matrix relative to a basis is the trace matrix, whose determinant is the discriminant — see discriminants and integral bases.
Ideal norms
The norm extends from elements to ideals, where it is defined as the index of the ideal in the maximal order and agrees with the element norm on principal ideals. See ideal norm computation.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.3. 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.