Splitting Fields and Algebraic Closures
Handbook guide to splitting fields and algebraic closures with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Splitting Fields
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.
Algebraic Closures
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.
Definition
If E is an extension of F and f ∈F[X], call f splits over E if f can be written as λ(X −α1) · · · (X −αk) for some α1, . . . , αk ∈E and λ ∈F. [There is a subtle point that should be mentioned. We would like to refer to the αi as “the” roots of f, but in doing so we are implicitly assuming that if β is an element of some extension E′ of E and f(β) = 0, then β must be one of the αi. This follows upon substituting β into the equation f(X) = λ(X −α1) · · · (X −αk) = 0.] If K is an extension of F and f ∈F[X], call K is a splitting field for f over F if f splits over K but not over any proper subfield of K containing F. Equivalently, K is a splitting field for f over F if f splits over K and K is generated over F by the roots α1, . . . , αk of f, in other words, F(α1, . . . , αk) = K.
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.
Proposition
If f ∈F[X] and deg f = n, then f has a splitting field K over F with [K : F] ≤n!.
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 α and β are roots of the irreducible polynomial f ∈F[X] in an extension E of F, then F(α) is isomorphic to F(β) via an isomorphism that carries α into β and is the identity on F.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Definition
If E and E′ are extensions of F and i is an isomorphism of E and E′, call i is an F-isomorphism if i fixes F, that is, i(a) = a for every a ∈F. Fhomomorphisms, F-monomorphisms, etc., are defined similarly.
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.
Isomorphism Extension Theorem Suppose that F and F ′ are isomorphic, and
Isomorphism Extension Theorem Suppose that F and F ′ are isomorphic, and the isomorphism i carries the polynomial f ∈F[X] to f ′ ∈F ′[X]. If K is a splitting field for f over F and K′ is a splitting field for f ′ over F ′, then i can be extended to an F-isomorphism of K and K′. In particular, if F = F ′ and i is the identity function, we conclude that any two splitting fields of f are F-isomorphic.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Example Find a splitting field for f(X) = X3 −2 over the rationals Q.
Find a splitting field for f(X) = X3 −2 over the rationals Q. If α is the positive cube root of 2, then the roots of f are α, α(−1 2 + i 1 2 √ 3) and α(−1 2 −i 1 2 √ 3). The polynomial f is irreducible, either by Eisenstein’s criterion or by the observation hat if f were factorable, it would have a linear factor, and there is no rational number whose cube is 2. Thus f is the minimal polynomial of α, so [Q(α) : Q] = 3.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Proposition
If C is a field, the following conditions are equivalent. (1) Every nonconstant polynomial f ∈C[X] has at least one root in C. (2) Every nonconstant polynomial f ∈C[X] splits over C. (3) Every irreducible polynomial f ∈C[X] is linear.
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
An extension C of F is an algebraic closure of F if C is algebraic over F and C is algebraically closed. Note that C is minimal among algebraically closed extensions of F. For if F ≤K ≤C and α ∈C, α /∈K, then since α is algebraic over F it is algebraic over K. But since α /∈K, the minimal polynomial of α over K is a nonlinear irreducible polynomial in K[X].
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.
Proposition
If E is generated over F by finitely many elements α1, . . . , αn algebraic over F (so that E = F(α1, . . . , αn)), then E is a finite 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.
Corollary
If E is an extension of F and A is the set of all elements in E that are algebraic over F (the algebraic closure of F in E), then A is a subfield of E.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Corollary (Transitivity of Algebraic Extensions) If E is algebraic over K (in
(Transitivity of Algebraic Extensions) If E is algebraic over K (in other words, every element of E is algebraic over K), and K is algebraic over F, then E is algebraic over 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.
Proposition
Let C be an algebraic extension of F. Then C is an algebraic closure of F if and only if every nonconstant polynomial in F[X] splits over C.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Theorem
Every field F has an algebraic closure. Informal argument. Well-order F[X] and use transfinite induction, beginning with the field F0 = F. At stage f we adjoin all roots of the polynomial f by constructing a splitting field for f over the field F<f that has been generated so far by the recursive procedure.
Proof / verification strategy: Partition the finite set into cosets or orbits, compare cardinalities, and use divisibility or stabiliser information to obtain the structural conclusion.
Theorem Any two algebraic closures C and C′ of F are F-isomorphic.
Any two algebraic closures C and C′ of F are F-isomorphic.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Quick-reference relationships
Problem-solving workflow
Name the base field and extension
Keep the direction of the extension and any intermediate fields explicit.
Classify the elements involved
Determine whether elements are algebraic, separable, normal, transcendental or generators of the extension.
Use minimal or splitting polynomials
Polynomial factorisation and root structure determine the relevant field construction.
Track extension degree
Apply basis arguments and degree multiplicativity before making claims about possible intermediate fields.
Relate automorphisms to fixed fields
For finite Galois situations, use the subgroup-field correspondence only after the extension hypotheses are satisfied.
Verify by root action
Represent automorphisms through their action on roots and check that all defining algebraic relations are preserved.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test kernel, ideal, quotient, polynomial, root, degree, basis, field. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- 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.
- Treating every automorphism of an extension as arbitrary on generators; algebraic relations must be preserved.
- Using the subgroup-field correspondence outside the finite Galois setting.
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 |
|---|---|---|
| 3.2 | Splitting Fields | 55–56 |
| 3.3 | Algebraic Closures | 57–58 |
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.