Number Fields and Algebraic Numbers
Field Isomorphism and the Normal Closure
Testing whether two number fields are isomorphic, computing the isomorphisms, and constructing the normal closure.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0582
Deciding whether two defining polynomials describe the same field, and constructing the maps between them, is required for table lookup, for Galois theory, and for combining results computed in different presentations.
The isomorphism test
Two fields of the same degree are isomorphic exactly when the defining polynomial of one has a root in the other.
Testing isomorphism
- Compare degreesDifferent degrees means not isomorphic.
- Compare discriminantsDifferent field discriminants means not isomorphic. A fast necessary condition.
- FactorFactor the first defining polynomial over the second field — see factoring over number fields.
- Read the rootsEach linear factor gives an isomorphism; the root is the image of the generator.
Automorphisms
Applying the test to a field against itself gives its automorphism group: the roots of the defining polynomial that lie in the field are exactly the images of the generator under automorphisms.
| Number of roots in the field | Conclusion |
|---|---|
| 1 (the generator itself) | Only the identity automorphism |
| All n | The field is Galois over the rationals |
| Between | Partially normal; the count is the automorphism group order |
The normal closure
The normal closure is the smallest Galois extension containing the field — the splitting field of the defining polynomial. It is constructed by successively adjoining roots.
Constructing the normal closure
- Factor over the fieldSplit off the known root.
- Adjoin a root of a remaining factorProducing a larger field.
- RepeatUntil the polynomial splits completely.
- ReduceApply polynomial reduction at each stage to control growth.
Practical guidance
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.5.3. 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.