Engineering / Mathematics — Fields, Series and Factorisation
General Properties of Extension Fields
Field extensions, degree, algebraic elements and minimal polynomials.
Executive summary
An extension field contains a smaller field and is a vector space over it. The dimension is the degree, and it controls the whole theory through the tower law.
An element is algebraic when it satisfies a polynomial over the base field, and its minimal polynomial is the fundamental invariant attached to it.
Learning objectives
- Define extensions, degree and algebraic elements.
- Construct simple extensions as quotient algebras.
- Apply the tower law.
01Degree and the tower law
Extension and degree
If K ⊆ L are fields, L is an extension of K and is a vector space over K. Its dimension is the degree [L : K].
Tower law
For K ⊆ L ⊆ M with finite degrees, [M : K] = [M : L] · [L : K].
The multiplicativity is what constrains the subfield structure of finite fields: F_{p^d} sits inside F_{p^k} exactly when d divides k, since the degrees must multiply to k.
[F_{p^k} : F_p] = k, so subfields correspond exactly to divisors of k02Algebraic elements and minimal polynomials
Algebraic element and minimal polynomial
α ∈ L is algebraic over K if it is a root of some non-zero polynomial in K[X].
Its minimal polynomial is the monic generator of the ideal of polynomials vanishing at α. It is irreducible.
Simple extensions
If α is algebraic over K with minimal polynomial m of degree n, then
K(α) ≅ K[X]/(m) and [K(α) : K] = n.
The isomorphism is the evaluation map sending X to α, whose kernel is the ideal generated by m. Irreducibility of m is what makes the quotient a field, and the first isomorphism theorem does the rest.
03Finite versus algebraic
| Property | Meaning | Relationship |
|---|---|---|
| Finite extension | Finite degree | Implies algebraic |
| Algebraic extension | Every element algebraic | Does not imply finite |
| Simple extension | Generated by one element | Finite iff that element is algebraic |
| Transcendental element | Satisfies no polynomial | Generates an infinite extension |
Every finite extension is algebraic, since the powers of any element must eventually become linearly dependent in a finite-dimensional space, giving a polynomial relation. The converse fails — the algebraic closure of the rationals is algebraic but infinite over them.
For finite fields the distinction does not arise: every extension of a finite field by a finite field is finite and algebraic, and every element satisfies the polynomial X^{q^k} − X.
04Frequently asked questions
Why is the minimal polynomial irreducible?
Because a factorisation would give a product vanishing at α, and in a field one factor must vanish there — contradicting minimality of the degree. Irreducibility is forced rather than assumed.
Does every polynomial have a root somewhere?
Yes. For irreducible f over K, the quotient K[X]/(f) is a field containing a root, namely the class of X. Iterating gives a splitting field in which f factors completely.
Are all extensions of the same degree isomorphic?
For finite fields, yes — any two fields of the same order are isomorphic. In general no: Q(√2) and Q(√3) both have degree 2 over Q and are not isomorphic as extensions.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 376-378.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.