Number Fields and Algebraic Numbers
Algebraic Numbers and Minimal Polynomials
Algebraic numbers, minimal polynomials, algebraic integers, and the computational tests that distinguish them.
Engineering / MathematicsNumber Fields and Algebraic Numbers2 min readKV-MATH-0573
An algebraic number is a root of a non-zero polynomial with rational coefficients. The minimal polynomial is the canonical object attached to it, and nearly every computation begins by obtaining one.
Minimal polynomial
- Definition
- The monic rational polynomial of least degree having the number as a root. It is unique and irreducible.
- Degree
- The degree of the algebraic number, and the dimension of the field it generates over the rationals.
- Conjugates
- The other roots of the minimal polynomial. There are exactly as many as the degree, all distinct.
Algebraic integers
An algebraic integer is an algebraic number whose minimal polynomial has integer coefficients. Since the minimal polynomial is monic by definition, this is a genuine restriction.
Closure properties
Algebraic numbers form a field and algebraic integers form a ring. Both closure statements are constructive: minimal polynomials of sums and products are computed by resultants.
| Operation | Construction |
|---|---|
| Sum | Resultant eliminating one variable from the two minimal polynomials |
| Product | Similar resultant construction |
| Inverse | Reverse the coefficient list of the minimal polynomial |
| Conjugates | Other roots of the same minimal polynomial |
Obtaining a minimal polynomial
From a matrix representation, the minimal polynomial is extracted from the characteristic polynomial by squarefree factorisation — see the Hessenberg method. From a numerical approximation, it can be recovered by lattice-based dependence detection, with mandatory exact verification.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.1.1. 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.