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Quartic Galois Groups and Resolvent Cubics

Handbook guide to quartic galois groups and resolvent cubics with core definitions, structural results, reasoning methods and verification checks.

Approx. 10 min read

Handbook scope. This handbook article develops quartic galois groups and resolvent cubics as a connected part of abstract algebra. The supplied source treats the topic through the sequence Quartic extensions and resolvent cubic appendix. The presentation below preserves that dependency: definitions come first, then structural results, constructions and calculation methods, followed by checks that expose the hypotheses most likely to be missed. Proofs from the source are condensed to proof strategies rather than reproduced line-for-line, while theorem statements, algebraic relationships and decision conditions are retained in technical form.

Section A6: pp. 128–130

1source section integrated

7formal results and definitions distilled

3source pages in the primary theory range

How the topic fits together

Quartic extensions and resolvent cubic appendix

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.

Lemma · A6.7

Lemma F(V ∩G) = F(u, v, w), where

F(V ∩G) = F(u, v, w), where u = (x1 + x2)(x3 + x4), v = (x1 + x3)(x2 + x4), w = (x1 + x4)(x2 + x3).

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.

Definition · A6.8

Definition

The resolvent cubic of f(X) = X4 + qX2 + rX + s is g(X) = (X −u)(X −v)(X −w). To compute g, we must express its coefficients in terms of q, r and s. First note that u −v = −(x1 − x4)(x2 −x3), u −w = −(x1 −x3)(x2 −x4), v −w = −(x1 −x2)(x3 −x4). Thus f and g have the same discriminant.

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.

Result · A6.9

An Explicit Formula For The Resolvent Cubic:

An Explicit Formula For The Resolvent Cubic: g(X) = X3 −2qX2 + (q2 −4s)X + r2. We need some results concerning subgroups of Sn, n ≥3.

Lemma · A6.10

Lemma

(i) An is generated by 3-cycles, and every 3-cycle is a commutator. (ii) The only subgroup of Sn with index 2 is An.

Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.

Proposition · A6.11

Proposition

Let G be a subgroup of S4 whose order is a multiple of 4, and let V be the four group (see the discussion preceding A6.7). Let m be the order of the quotient group G/(G ∩V ). Then (a) If m = 6, then G = S4; (b) If m = 3, then G = A4; (c) If m = 1, then G = V ; (d) If m = 2, then G = D8 or Z4 or V ; (e) If G acts transitively on {1, 2,3, 4}, then the case G = V is excluded in (d). [In all cases, equality is up to isomorphism.]

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.

Theorem · A6.12

Theorem

Let f be an irreducible separable quartic, with Galois group G. Let m be the order of the Galois group of the resolvent cubic. Then: (a) If m = 6, then G = S4; (b) If m = 3, then G = A4; (c) If m = 1, then G = V ; (d) If m = 2 and f is irreducible over L = F(u, v, w), where u, v and w are the roots of the resolvent cubic, then G = D8; (e) If m = 2 and f is reducible over L, then G = Z4.

Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.

Example · A6.13

Example

Let f(X) = X4 + 3X2 + 2X + 1 over Q, with q = 3, r = 2, s = 1. The resolvent cubic is, by (A6.9), g(X) = X3 −6X2 + 5X + 4. To calculate the discriminant of g, one can use the general formula in (A6.6), or compute g(X + 2) = (X + 2)3 −6(X + 2)2 + 5(X + 2) + 4 = X3 −7X −2. [The rational root test gives irreducibility of g and restricts a factorization of f to (X2 + aX ± 1)(X2 −aX ± 1), a ∈Z, which is impossible. Thus f is irreducible as well.] One has D(g) = −4(−7)3 −27(−2)2 = 1264, which is not a square in Q.

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

F(V ∩G) = F(u, v, w), where u = (x1 + x2)(x3 + x4), v = (x1 + x3)(x2 + x4), w = (x1 + x4)(x2 + x3).

The resolvent cubic of f(X) = X4 + qX2 + rX + s is g(X) = (X −u)(X −v)(X −w).

First note that u −v = −(x1 − x4)(x2 −x3), u −w = −(x1 −x3)(x2 −x4), v −w = −(x1 −x2)(x3 −x4).

An Explicit Formula For The Resolvent Cubic: g(X) = X3 −2qX2 + (q2 −4s)X + r2.

Then (a) If m = 6, then G = S4;

(b) If m = 3, then G = A4;

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 source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.

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
A6	Quartic extensions and resolvent cubic appendix	128–130

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.