Number Fields and Algebraic Numbers
Number Fields: Definition and Basic Properties
Number fields as finite extensions of the rationals, their embeddings and signature, and the presentation on which all computation depends.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0574
A number field is a finite extension of the rationals. Every one is generated by a single element, which is what makes them computationally tractable: the whole field is described by one irreducible polynomial.
Presentation
Embeddings and signature
A field of degree n has exactly n embeddings into the complex numbers, corresponding to the roots of the defining polynomial. Real roots give real embeddings; complex roots pair into conjugate embeddings.
Signature
The presentation is not unique
The same field admits many defining polynomials, and they differ enormously in how pleasant they are to compute with.
| Property of the polynomial | Consequence |
|---|---|
| Large coefficients | Large intermediate values throughout |
| Large discriminant | Expensive maximal order computation, since the discriminant must be factored |
| Large index in the maximal order | Many primes require the hard decomposition case |
| Small, reduced | Everything downstream is cheaper |
What is computed about a field
- The maximal order, via Round 2.
- The field discriminant, which measures ramification.
- Decomposition of primes, via decomposition theory.
- The class group and unit group — see the four core tasks.
- The Galois group, via resolvents.
Relative extensions
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.1.2. 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.