Cyclotomic Fields and Cubic Galois Groups
Handbook guide to cyclotomic fields and cubic galois groups with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Cyclotomic 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.
The Galois Group of a Cubic
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
Cyclotomic extensions of a field F are formed by adjoining nth roots of unity. Formally, a cyclotomic extension of F is a splitting field E of f(X) = Xn −1 over F . The roots of f are called nth roots of unity, and they form a multiplicative subgroup of the group E∗of nonzero elements of E. This subgroup must be cyclic by (6.4.4).
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.
Definition
The nth cyclotomic polynomial is defined by Ψn(X) = Y i (X −ωi) where the ωi are the primitive nth roots of unity in the field C of complex numbers. Thus the degree of Ψn(X) is ϕ(n). From the definition, one has Ψ1(X) = X−1 and Ψ2(X) = X +1. In general, the cyclotomic polynomials can be calculated by the following recursion formula, in which d runs through all positive divisors of n.
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
Xn −1 = Y d|n Ψd(X). In particular, if p is prime, then Ψp(X) = Xp −1 X −1 = Xp−1 + Xp−2 + · · · + X + 1.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Corollary
The Galois group G of the nthcyclotomic extension Q(ω)/Q is isomorphic to the group Un of units mod n.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Definition
Let f be a polynomial with distinct roots x1, . . . , xn as above. Define ∆(f) = Y i<j (xi −xj). The discriminant of f is defined by D(f) = ∆2 = Y i<j (xi −xj)2. Let’s look at a quadratic polynomial f(X) = X2 + bX + c, with roots 1 2(−b ± √ b2 −4c).
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 be a splitting field of the separable polynomial f over F, so that E/F is Galois. (a) D(f) belongs to the base field F. (b) Let σ be an automorphism in the Galois group G of f. Then σ is an even permutation (of the roots of f) iffσ(∆) = ∆, and σ is odd iffσ(∆) = −∆.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Result
The Galois Group of a Cubic In the appendix to Chapter 6, it is shown that the discriminant of the abbreviated cubic X3+pX +q is −4p3−27q2, and the discriminant of the general cubic X3+aX2+bX +c is a2(b2 −4ac) −4b3 −27c2 + 18abc. Alternatively, the change of variable Y = X + a 3 eliminates the quadratic term without changing the discriminant.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Example
Let f(X) = X3 −31X + 62 over Q. An application of the rational root test (Section 2.9, Problem 1) shows that f is irreducible. The discriminant is −4(−31)3 −27(62)2 = 119164 −103788 = 15376 = (124)2, which is a square in Q. Thus the Galois group of f is A3.
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.
Some Generating Sets of Sn
Some Generating Sets of Sn (i) Sn is generated by the transpositions (1,2), (1,3),. . ., (1,n). [An arbitrary transposition (i, j) can be written as (1, i)(1, j)(1, i).] (ii) Sn is generated by transpositions of adjacent digits, i.e., (1, 2),(2,3), . . . ,(n −1, n). [Since (1, j −1)(j −1, j)(1, j −1) = (1, j), one has (1,2)(2,3)(1,2)=(1,3), (1,3)(3,4)(1,3)=(1,4), etc., and the result follows from (i).] (iii) Sn is generated by the two permutations σ1 = (1, 2) and τ = (1, 2, . . . , n). (iv) Sn is generated by (1,2) and (2,3,. . . ,n). [(1,2)(2,3,. . . ,n)=(1,2,3,. . . ,n), and (iii) applies.]
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Lemma
If f is an irreducible separable polynomial over F of degree n, and G is the Galois group of f, then n divides |G|. If n is a prime number p, then G contains a p-cycle.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
If f is irreducible over Q and of prime degree p, and f has exactly two nonreal roots in the complex field C, then the Galois group G of f is Sp.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test polynomial, root, degree, field, automorphism, exact, prime, Ext. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
- 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 |
|---|---|---|
| 6.5 | Cyclotomic Fields | 113–115 |
| 6.6 | The Galois Group of a Cubic | 116–117 |
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.