Software, Tables and Sources
Further Reading and Source Notes
How this collection is organised, how to read it, and notes on the source material and its treatment.
Engineering / MathematicsSoftware, Tables and Sources2 min readKV-MATH-0682
This page records how the collection is organised, how the material relates to its source, and where to go next.
How the collection is organised
The collection runs from arithmetic foundations through to the algorithms that depend on them. Earlier streams are prerequisites for later ones, and cross-references run in both directions.
| Layer | Streams |
|---|---|
| Foundations | Multiprecision arithmetic, Euclidean algorithms, finite fields |
| Structural tools | Linear algebra, normal forms, lattices and LLL |
| Polynomials | Arithmetic, GCD, factorisation |
| Number fields | Fields, orders, ideals, prime decomposition, maximal orders |
| Global invariants | Class groups, units, regulators, quadratic fields, Galois groups |
| Applications | Elliptic curves, primality proving, factoring |
Suggested pathways
Several routes through the material are set out in learning pathways. The quadratic field route is the most concrete: every general phenomenon appears there in a form small enough to compute by hand.
Source notes
Where the treatment is deliberately partial
- Proofs are omitted throughout; the emphasis is on what an algorithm does, what it assumes, what it costs and how it fails.
- Complexity statements are given in the form used in practice, with heuristic content flagged where it exists.
- Numerical parameter tables are pointed to rather than reproduced, since they change and are better taken from a maintained source.
- Class field theory, higher-genus curves and modular forms are referenced only where an algorithm depends on them.
Going further
Depth in number fields
The maximal order, decomposition and class group streams point to the relative and class field theory material that extends them.
Depth in curves
The elliptic curve stream connects to modular forms, isogeny graphs and the arithmetic of higher genus.
Depth in factoring
The number field sieve rewards close study; see polynomial selection.
Practice
Working through computations in a real system is the most effective next step — see software packages.
Using this collection
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — collection orientation material. 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.