Number Fields and Algebraic Numbers
The Subfield Problem
Finding the subfields of a number field, by lattice methods and by linear algebra over the complex numbers.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0581
Determining the subfields of a number field is needed for Galois group computation, for simplifying arithmetic, and for understanding the field's structure. Two approaches are standard and they have quite different characters.
The criterion
An element generates a proper subfield exactly when its minimal polynomial has degree properly dividing the field degree — equivalently, when its characteristic polynomial is a proper power.
The lattice approach
Subfield generators are, in a suitable sense, small: they satisfy a polynomial of low degree, which constrains them. Reducing the lattice of the maximal order and testing the short vectors finds them.
Subfields by lattice reduction
- Reduce the order latticeWith respect to the T2 form.
- Test short elementsCompute characteristic polynomials and check for proper powers.
- AssembleCollect the subfields found and remove duplicates by comparing canonical polynomials.
The linear algebra approach
Subfields correspond to partitions of the embeddings into blocks that are compatible with the field structure. Working with numerical embeddings, candidate block systems are enumerated and each is tested by constructing the corresponding subfield generator.
| Approach | Character | Completeness |
|---|---|---|
| Lattice reduction | Fast, heuristic | May miss subfields |
| Block systems over C | Systematic, numerical | Complete if the enumeration is exhaustive |
| Via the Galois group | Exact once the group is known | Complete; needs the group first |
Relation to Galois theory
For a Galois extension the subfields correspond exactly to subgroups of the Galois group, so computing the group solves the subfield problem completely. For a non-Galois field the correspondence is with subgroups of the group of the normal closure containing a fixed point stabiliser. See the resolvent method.
Applications
Subfields allow arithmetic to be performed in a smaller field where possible, permit relative presentations, and are used in factoring over number fields to reduce degree.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.5.1-4.5.2. 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.