Separable and Normal Field Extensions
Handbook guide to separable and normal field extensions with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Separability
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.
Normal 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.
Theorem
If E is an algebraic extension of F, C is an algebraic closure of F, and i is an embedding (that is, a monomorphism) of F into C, then i can be extended to an embedding of E into C. Informal argument. Each α ∈E is a root of some polynomial in F[X], so if we allow α to range over all of E, we get a collection S of polynomials in F[X]. Within C, carry out the recursive procedure of (3.3.7) on the polynomials in S.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Definition
An irreducible polynomial f ∈F[X] is separable if f has no repeated roots in a splitting field; otherwise f is inseparable. If f is an arbitrary polynomial, not necessarily irreducible, then we call f separable if each of its irreducible factors is separable. Thus if f(X) = (X −1)2(X −3) over Q, then f is separable, because the irreducible factors (X −1) and (X −3) do not have repeated roots. We will see shortly that over a field of characteristic 0 (for example, the rationals), every polynomial is separable.
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(X) = a0 + a1X + · · · + anXn ∈F[X], let f ′ be the derivative of f, defined by f ′(X) = a1 + 2a2X + · · · + nanXn−1. [Note that the derivative is a purely formal expression; we completely ignore questions about existence of limits. One can check by brute force that the usual rules for differentiating a sum and product apply]. If g is the greatest common divisor of f and f ′, then f has a repeated root in a splitting field if and only if the degree of g is at least 1.
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.
Corollary
(1) Over a field of characteristic zero, every polynomial is separable. (2) Over a field F of prime characteristic p, the irreducible polynomial f is inseparable if and only if f ′ is the zero polynomial. Equivalently, f is a polynomial in Xp; we abbreviate this as f ∈F[Xp].
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.
The Frobenius Automorphism
The Frobenius Automorphism Let F be a finite field of characteristic p, and define f : F →F by f(α) = αp. Then f is an automorphism. In particular, if α ∈F then α = βp for some β ∈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.
Proposition Over a finite field, every polynomial is separable.
Over a finite field, every polynomial is separable.
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.
Definition
If E is an extension of F and α ∈E, then α is separable over F if α is algebraic over F and min(α, F) is a separable polynomial. If every element of E is separable over F, call E is a separable extension of F or the extension E/F is separable or E is separable over F. By (3.4.3) and (3.4.5), every algebraic extension of a field of characteristic zero or a finite field is separable.
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
If F ≤K ≤E and E is separable over F, then K is separable over F and E is separable over K.
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.
Example
Let F = Fp(t) be the set of rational functions (in the indeterminate t) with coefficients in the field with p elements (the integers mod p). Thus an element of F looks like a0 + a1t + · · · + amtm b0 + b1t + · · · + bntn . with the ai and bj in Fp. Adjoin p√ t, that is, a root of Xp −t, to create the extension E. Note that Xp −t is irreducible by Eisenstein, because t is irreducible in Fp[t].
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.
Lemma
Let σ : E →E be an F-monomorphism, and assume that the polynomial f ∈F[X] splits over E. If α is a root of f in E, then so is σ(α). Thus σ permutes the roots of 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.
Theorem
Let E/F be a finite separable extension of degree n, and let σ be an embedding of F in C. Then σ extends to exactly n embeddings of E in C, in other words, there are exactly n embeddings τ of E in C such that the restriction τ|F of τ to F coincides with σ. In particular, taking σ to be the identity function on F, there are exactly n Fmonomorphisms of E into 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.
Example
Adjoin the positive cube root of 2 to the rationals to get E = Q( 3√ 2). The roots of the irreducible polynomial f(X) = X3 −2 are 3√ 2, ω 3√ 2 and ω2 3√ 2, where ω = ei2π/3 = −1 2 + 1 2 √ 3 and ω2 = ei4π/3 = −1 2 −1 2 √ 3. Notice that the polynomial f has a root in E but does not split in E (because the other two roots are complex and E consists entirely of real numbers). We give a special name to extensions that do not have this annoying drawback.
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.
Definition
The algebraic extension E/F is normal (we also say that E is normal over F) if every irreducible polynomial over F that has at least one root in E splits over E. Equivalently, if α ∈E, then all conjugates of α over F (i.e., all roots of the minimal polynomial of α over F) belong to E. Here is an equivalent condition. (The hypothesis that E/F is finite rather than simply algebraic can be removed, but we will not need the more general result.)
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
The finite extension E/F is normal if and only if every F-monomorphism of E into an algebraic closure C is actually an F-automorphism 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.
Theorem
The finite extension E/F is normal if and only if E is a splitting field for some polynomial f ∈F[X].
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
Let F ≤K ≤E, where E is a finite extension of F. If E/F is normal, so is E/K.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Definitions and Comments
If E/F is normal and separable, it is called a Galois extension; we also say that E is Galois over F. It follows from (3.5.2) and (3.5.5) that if E/F is a finite Galois extension, then there are exactly [E : F] F-automorphisms of E. If E/F is finite and separable but not normal, then at least one F-embedding of E into an algebraic closure must fail to be an automorphism of E. Thus in this case, the number of F-automorphisms of E is less than the degree of the extension.
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.
Example
Let E = Q( 3√ 2), as in (3.5.3). The Galois group of the extension consists of the identity automorphism alone. For any Q-monomorphism σ of E must take 3√ 2 into a root of X3 −2. Since the other two roots are complex and do not belong to E, 3√ 2 must map to itself.
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.
The Normal Closure Let E be a finite extension of F, say E = F(α1, . . . , αn). If
The Normal Closure Let E be a finite extension of F, say E = F(α1, . . . , αn). If N ⊇E is any normal extension of F, then N must contain the αi along with all conjugates of the αi, that is, all roots of min(αi, F), i = 1, . . . , n. Thus if f is the product of these minimal polynomials, then N must contain the splitting field K for f over F. But K/F is normal by (3.5.7), so K must be the smallest normal extension of F that contains 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.
Theorem of the Primitive Element If E/F is a finite separable extension, then
of the Primitive Element If E/F is a finite separable extension, then E = F(α) for some α ∈E. Call α is a primitive element of E 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.
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 polynomial, root, degree, basis, field, injective, Ext. These checks are used here as verification themes rather than copied as answer text.
The supplied worked solutions for this section repeatedly test polynomial, root, degree, basis, field, automorphism, exact, prime. 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.
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
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 |
|---|---|---|
| 3.4 | Separability | 59–61 |
| 3.5 | Normal Extensions | 62–64 |
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.