Teaching objectives
Last group theory lab, for first-year university. It follows “Groups you can see”, “Cosets and quotients”, “Homomorphisms and automorphisms” and “Centre, commutators and solvable groups”, and uses what is learnt there (permutations, commutators, solvable groups) to prove that there is no formula in radicals for the general equation of degree five.
The proof is not Galois’s one with fields. It is V. I. Arnold’s topological proof: move a coefficient of the polynomial along a closed loop in the complex plane and watch what the roots do. Everything is on screen: two planes, that of the coefficient a, with its dangerous points in red, and that of the roots, with five coloured points leaving trails. When the loop closes, the roots are back in the same places… but in a different order.
What is learnt
- Writing formulas as chains of radicals, and what “solvable in radicals” means exactly.
- Seeing the roots of a polynomial as points that move when a coefficient changes, and finding the values where they collide (branch points) from P = P′ = 0.
- Attaching a permutation of the roots to every loop, composing loops and seeing that for x⁵ − x + a every permutation can be obtained.
- Seeing that the permutations of the loops form a group and recognising it (Zₙ, Δₙ, Aₙ or Sₙ) from the full list: S₅ for x⁵ − x + t, but Z₅ or Δ₅ for other quintics.
- Reading permutations as arrow diagrams, multiplying and inverting them, and finding their parity by counting transpositions.
- Understanding why a formula in radicals cannot follow the roots along commutators of loops, and commutators of commutators.
- Seeing that in S₄ nested commutators run out (which is why degree 4 has a formula) while in A₅ they never do.
- With polynomials chosen by the student: drawing one’s own loops, computing the monodromy group and deciding whether that particular equation has a formula. There are presets with groups Δ₅, S₅, Z₅, S₄, Δ₄, Z₄ and Δ₃.
Key mathematical ideas
- A loop permutes the roots. The roots of x⁵ − x + a collide only at four values of a, ±4/(5·⁴√5) and ±4i/(5·⁴√5). Going around one of them swaps two roots.
- Radicals commute. A loop multiplies all values of ᵏ√p(a) by the same root of unity, so a commutator of loops leaves them in place; with N nested radicals, N-fold nested commutators do.
- A₅ is made of commutators. Every even permutation of five elements is a product of commutators of even permutations, so there is always a loop the formula cannot follow.
- Monodromy = Galois. The group generated by the loops is the Galois group of the polynomial over C(t): there is a formula if and only if it is solvable. Some particular quintics, such as Chebyshev’s x⁵ − 5x³ + 5x + t (group Δ₅), do have one.
Sources and credits
Follows lecture 5, “Equations of degree five”, of Mathematical Omnibus by D. Fuchs and S. Tabachnikov (AMS, 2007), based on V. I. Arnold’s lectures for Moscow schoolchildren (1963–64), and the book Abel’s Theorem in Problems and Solutions by V. B. Alekseev, which comes from the same lectures.
Classroom notes
- “Going around a₀” is the key room: project it and run the loop slowly so that everyone sees the two roots cross.
- “The loop explorer” and “Loops by hand” work well in pairs: one proposes a permutation and the other draws the loop.
- “One radical is not enough” is the heart of the proof; it is worth pausing on the computation m₁ + m₂ − m₁ − m₂ = 0.
- “The monodromy group” keeps a surprise for the end: some quintics do have a formula.
Room-by-room contents
Room 1 · Formulas as chains
The formulas for degrees 2 and 3 written as chains of radicals (x₁² = p² − 4q, …). Definition of “solvable in radicals”.
Student tasks
- Run Cardano’s chain for x³ − 6x − 9 = 0.
- Count the values of a chain with exponents 2, 3 and 5.
Room 2 · Roots of unity
Multiplying complex numbers adds angles; xᵏ = c has k solutions x₁εʲ. A draggable point z with its powers.
Student tasks
- Take z to a fifth root of 1.
- Angles and number of solutions.
Room 3 · An easy quintic
x⁵ + a: one turn of a turns the roots by 1/5; loops only give rotations and the formula ⁵√−a exists.
Student tasks
- Count turns and read the permutation of a loop.
Room 4 · Roots that move
Drag a in x⁵ − x + a and the five roots move without colliding.
Student tasks
- Bring two roots close (discovery).
- Which pair approaches along the real axis.
Room 5 · The dangerous values
Double root ⇔ 5b⁴ = 1 ⇒ a⁴ = 4⁴/5⁵: four values ±a₀, ±ia₀ (a₀ ≈ 0.535).
Student tasks
- Why a double root makes the derivative vanish.
- Drag a to −a₀ and watch two roots collide; where the double root is.
- Why there are four and why they form a square.
Room 6 · A closed loop
A loop from 0 brings the roots back to the set, but permuted; around nothing, the identity.
Student tasks
- Draw a loop that does not permute and one that does.
Room 7 · Going around a₀
Near b₀, a ≈ a₀ − 10b₀³ε²: one turn of a is half a turn of x and 0 and 1 swap.
Student tasks
- Which roots swap; two turns.
Room 8 · Turning by 90°
The roots of x⁵ − x + ia are i times those of x⁵ − x + a: the other three loops give (2 5), (3 5), (4 5).
Student tasks
- Predict each swap before seeing it.
Room 9 · Permutations with arrows
Cycle notation, arrow diagrams, product by stacking (α first) and inverse by reflecting.
Student tasks
- Read a diagram in cycles; products (1 2)(3 4) · (2 3)(4 5) in both orders; inverse of (1 4 3 5 2).
Room 10 · Transpositions and parity
A cycle of length k is a product of k − 1 transpositions; parity read off disjoint cycles; 60 even and 60 odd; a commutator is always even.
Student tasks
- Count transpositions and decide parities.
Room 11 · Composing loops
Composing loops multiplies permutations; the basic loops (i 5) give any permutation.
Student tasks
- Build words of loops for five target permutations.
Room 12 · Branch points
Double root ⇔ P = P′ = 0; with t the constant term, Q′(b) = 0 and t = −Q(b). Cases x³ − 3x + t, x⁴ − 2x² + t (two double roots at t = 1) and x⁵ + t.
Student tasks
- Compute and count branch points.
- Read the loop of a double double root.
Room 13 · Each polynomial, its group
The permutations the loops produce form a group, the monodromy group. The engine gives the full list, sorted by cycle type, and the student recognises the group among Zₙ, Δₙ, Aₙ and Sₙ. Worked example: x⁵ − x + t, with all 120 permutations (S₅).
Student tasks
- Recognise Z₅ (x⁵ + t), S₃ (x³ − 3x + t), Δ₄ (x⁴ − 2x² + t), Δ₅ (Chebyshev) and S₄ (x⁴ + tx + 1).
- Rule out options of the same order by their structure, such as Z₆, Z₈ or Z₁₀.
Room 14 · Loops by hand
Only drawn loops: you see each root follow its trail and which seat it ends in, and the permutation with its diagram.
Student tasks
- Draw loops giving (1 5), a 3-cycle, no fixed root, the identity around a point and 1 → 3 in x⁵ + t.
Room 15 · The big loop
The loop around all four points gives a 5-cycle; ℓ₁ℓ₂ℓ₃ℓ₄ = (1 2 3 4 5).
Student tasks
- Compute the product and read the big loop.
Room 16 · Commutators of loops
[ℓ₁, ℓ₂] = ℓ₁ℓ₂ℓ₁⁻¹ℓ₂⁻¹ gives the commutator of the permutations: (1 5 2), (1 5 3), (2 5 3).
Student tasks
- Compute three commutators.
Room 17 · One radical is not enough
A loop multiplies all values of ᵏ√p(a) by the same εᵐ; in a commutator m₁ + m₂ − m₁ − m₂ = 0.
Student tasks
- Follow exponents; conclude that one radical cannot solve the quintic.
Room 18 · The tower of radicals
Tower a → x₁ → … → x_N; N-fold nested commutators bring every level back (lemmas 5.6-5.7).
Student tasks
- Count values, levels and pieces.
Room 19 · Why degree 4 works
In S₄ commutators of even permutations give only V₄; the cubic x³ + ax − 1.
Student tasks
- Find the 4 permutations.
- Dangerous values of the cubic: how many and why they form an equilateral triangle.
Room 20 · A₅ is made of commutators
Every even permutation is a product of 3-cycles and double transpositions; theorem: with 5 elements both are commutators of even permutations, (1 2 3) = [(1 2)(4 5), (1 2 3)] and (1 2)(3 4) = [(2 3 4), (1 2 4)]. A calculator for [α, β].
Student tasks
- Commutator of two commuting permutations; how [α, β] and [β, α] are related.
- Work out (1 2)(2 3); what the (4 5) is for.
- Count the commutators of A₅ and A₄.
Room 21 · The nested loop
[[ℓ₁, ℓ₂], [ℓ₂, ℓ₃]] gives (1 5)(2 3), with 16 basic pieces.
Student tasks
- Read its permutation and count pieces.
Room 22 · A series that never goes down
A₅′ = A₅: there is always an N-fold nested loop that moves the roots; S₄ does go down.
Student tasks
- Sizes in the derived series.
Room 23 · The loop explorer
Polynomial of your choice (presets Δ₅, S₅, Z₅, S₄, Δ₄, Z₄, Δ₃ or free, with t in any coefficient); the engine draws the branch points; loops by hand and ready-made; monodromy group in free play.
Student tasks
- Six challenges on the presets.
- Free play with the group shown.
Room 24 · The monodromy group
The basic loops generate the Galois group; solvable ⇔ there is a formula.
Student tasks
- Identify Δ₅, S₄, Z₄, Δ₄ and S₅.
- Which quintic has a formula.
Room 25 · Your quintic with a formula
Discovery: build a quintic with a solvable group other than x⁵ + t.
Student tasks
- Find one (hint after 3 tries, way out after 6).
Room 26 · The whole proof
The seven steps of section 5.10.
Student tasks
- Order them, find the false step, what fails in degree 4.
Room 27 · Degree 5 and beyond
xⁿ − xⁿ⁻⁴ + axⁿ⁻⁵ = xⁿ⁻⁵(x⁵ − x + a); a sober history: Ruffini, Abel, Galois, Arnold.
Student tasks
- Four closing questions.
Rooms to project
The most striking ones to show and discuss in class.