Cyclic Extensions, Kummer Extensions and Solvability by Radicals
Cyclic Extensions, Kummer Extensions and Solvability by Radicals: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Cyclic and Kummer 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.
Solvability By Radicals
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.
Assumptions, Comments and a Definition Assume
Assumptions, Comments and a Definition Assume (i) E is a splitting field for f(X) = Xn −a over F, where a ̸= 0. (ii) F contains a primitive nthroot of unity ω. These are natural assumption if we want to allow the computation of nthroots. If θ is any root of f in E, then the roots of f are θ, ωθ, . . . , ωn−1θ.
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
Under the assumptions of (6.7.1), E/F is a cyclic extension and the order of the Galois group G is a divisor of n. One has |G| = n if and only if f(X) is irreducible over 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.
Theorem
Let E/F be a cyclic extension of degree n, where F contains a primitive nthroot of unity ω. Then for some nonzero a ∈F, f(X) = Xn −a is irreducible over F and E is a splitting field for f over 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
A Kummer extension is a finite Galois extension with an abelian Galois group.
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
Let E/F be a finite extension, and assume that F contains a primitive nthroot of unity ω. Then E/F is a Kummer extension whose Galois group G has an exponent dividing n if and only if there are nonzero elements a1, . . . , ar ∈F such that E is a splitting field of (Xn −a1) · · · (Xn −ar) over F . [For short, E = F( n√a1, . . . , n√ar).]
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 We wish to solve the polynomial equation f(X) = 0, f ∈F[X], under
We wish to solve the polynomial equation f(X) = 0, f ∈F[X], under the restriction that we are only allowed to perform ordinary arithmetic operations (addition, subtraction, multiplication and division) on the coefficients, along with extraction of nthroots (for any n = 2, 3, .. .). A sequence of operations of this type gives rise to a sequence of extensions F ≤F(α1) ≤F(α1, α2) ≤· · · ≤F(α1, . . . , αr) = E where αn1 1 ∈F and αni i ∈F (α1, . . . , αi−1), i = 2, . . . , r. Equivalently, one has F = F0 ≤F1 ≤. . . ≤Fr = E where Fi = Fi−1(αi) and αni i ∈Fi−1, i = 1, . . . , r. Call E is a radical extension of 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.
Proposition
Let E/F be a radical extension, and let N be the normal closure of E over F. Then N/F is also a radical extension.
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
Preparation for the Main Theorem If F has characteristic 0, then a primitive nthroot of unity ω can be adjoined to F to reach an extension F (ω); see (6.5.1). If E is a radical extension of F and F = F0 ≤F1 ≤· · · ≤Fr = E, one can replace Fi by Fi(ω), i = 1, . . . , r, and E(ω) will be a radical extension of F. By (6.8.2), one can pass from E(ω) to its normal closure over F. Here is the statement we are driving at: Let f ∈F[X], where F has characteristic 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.
Theorem
Galois’ Solvability Theorem Let K be a splitting field for f over F , where F has characteristic 0. Then f is solvable by radicals if and only if the Galois group of K/F is solvable.
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
Let f(X) = X5 −10X4 + 2 over the rationals. The Galois group of f is S5, which is not solvable. (See Section 6.6, Problem 3 and Section 5.7, Problem 5.) Thus f is not solvable by radicals. There is a fundamental idea that needs to be emphasized.
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.
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 order, polynomial, root, field, exact, prime, factor, discriminant. These checks are used here as verification themes rather than copied as answer text.
The supplied worked solutions for this section repeatedly test order, subgroup, quotient, polynomial, root, degree, field, 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 |
|---|---|---|
| 6.7 | Cyclic and Kummer Extensions | 118–119 |
| 6.8 | Solvability By Radicals | 120–122 |
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.