Field Extensions, Minimal Polynomials and Extension Degree
Field Extensions, Minimal Polynomials and Extension Degree: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Field Extensions
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definitions If F and E are fields and F ⊆E, we say that E is an extension of F,
Definitions If F and E are fields and F ⊆E, call E is an extension of F, and we write F ≤E, or sometimes E/F. If E is an extension of F, then in particular E is an abelian group under addition, and one may multiply the “vector” x ∈E by the “scalar” λ ∈F, and the axioms of a vector space are satisfied. Thus if F ≤E, then E is a vector space over F. The dimension of this vector space is called the degree of the extension, written [E : F].
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Lemma
Let f : F →E be a homomorphism of fields, i.e., f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) (all a, b ∈F), and f(1F ) = 1E. Then f is a monomorphism.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Theorem
Let f be a nonconstant polynomial over the field F. Then there is an extension E/F and an element α ∈E such that f(α) = 0.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Proposition
Let f and g be polynomials over the field F. Then f and g are relatively prime if and only if f and g have no common root in any extension of F.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Corollary
If f and g are distinct monic irreducible polynomials over F, then f and g have no common roots in any extension of F.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Definition
If E is an extension of F, the element α ∈E is called algebraic over F is there is a nonconstant polynomial f ∈F[X] such that f(α) = 0; if α is not algebraic over F, it is called transcendental over F. If every element of E is algebraic over F, then E is called an algebraic extension of F. Suppose that α ∈E is algebraic over F, and let I be the set of all polynomials g over F such that g(α) = 0. If g1 and g2 belong to I, so does g1 ± g2, and if g ∈I and c ∈F[X], then cg ∈I.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Theorem
If α ∈E is algebraic over F and the minimal polynomial m(X) of α over F has degree n, then F(α) = F[α], the set of polynomials in α with coefficients in F. In fact, F[α] is the set Fn−1[α] of all polynomials of degree at most n −1 with coefficients in F, and 1, α, . . . , αn−1 form a basis for the vector space F[α] over the field F. Consequently, [F(α) : F] = n.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Lemma
Suppose that F ≤K ≤E, the elements αi, i ∈I, form a basis for E over K, and the elements βj, j ∈J, form a basis for K over F. (I and J need not be finite.) Then the products αiβj, i ∈I, j ∈J, form a basis for E over F.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
The Degree is Multiplicative If F ≤K ≤E, then [E : F] = [E : K][K : F]. In
The Degree is Multiplicative If F ≤K ≤E, then [E : F] = [E : K][K : F]. In particular, [E : F] is finite if and only if [E : K] and [K : F] are both finite.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Theorem
If E is a finite extension of F, then E is an algebraic extension of F.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Quick-reference relationships
Problem-solving workflow
Fix the ring hypotheses
Record commutativity, identity, zero-divisor assumptions and whether the ring is a domain, field, PID, UFD or Euclidean domain.
Translate element questions into ideal questions
Divisibility, kernels, quotients and maximality often become clearer when expressed through generated ideals.
Choose a universal construction
For quotients, fractions or polynomial evaluation, define the candidate map and prove it is well-defined.
Separate existence from uniqueness
Division, factorisation and decomposition results often require different arguments for the two directions.
Use the strongest justified structure
Do not use field division in a general ring or unique factorisation before its hypotheses have been established.
Check the result in a concrete ring
Integers, residue rings and polynomial rings provide useful sanity checks for the abstract statement.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Using cancellation in a ring that may contain zero divisors.
- Treating every irreducible element as prime without the needed domain hypothesis.
- Assuming every ideal is principal.
- Applying polynomial root counting without an integral-domain hypothesis.
- Assuming an algebraic extension is automatically normal or separable.
- Confusing the degree of a polynomial with the degree of an extension when the polynomial is not minimal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 3.1 | Field Extensions | 52–54 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.