Number Fields and Algebraic Numbers
The Matrix (Regular) Representation of Algebraic Numbers
Representing a field element by its multiplication matrix, and reading trace, norm and characteristic polynomial off that matrix.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0576
Every field element acts on the field by multiplication, and that action is a linear map. Its matrix carries all the invariants of the element and converts algebraic questions into linear algebra.
The construction
Fix a basis of the field. Multiply each basis element by the element in question and express the result in the basis; the coefficient vectors are the columns of the matrix.
Reading the invariants
| Invariant | From the matrix |
|---|---|
| Trace | Trace of the matrix |
| Norm | Determinant of the matrix |
| Characteristic polynomial | Characteristic polynomial of the matrix |
| Minimal polynomial | Minimal polynomial of the matrix |
| Degree of the element | Degree of the minimal polynomial; a proper divisor of n indicates a subfield |
Computation
Building the matrix costs n multiplications in the standard representation. The characteristic polynomial is then computed by Hessenberg reduction or modular methods.
Integrality and the maximal order
Relative to an integral basis, an element is an algebraic integer exactly when its multiplication matrix has integer entries. This gives an effective integrality test and is the criterion used inside the Round 2 algorithm when enlarging an order.
Ideals as modules
The matrix representation also underlies ideal arithmetic: an ideal is a module stable under multiplication by the order, and stability is checked by applying multiplication matrices to the module's basis. See module representation.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.2.3. 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.