Solvability Theory

Discover why some equations can't be solved by radicals - the unsolvable quintic

Solvability Theory: The Unsolvable Quintic

This is the culmination of Galois Theory - answering a question that puzzled mathematicians for centuries: why can't we solve the quintic? The answer connects solvable groups to radical extensions in one of the most beautiful theorems in mathematics. We'll discover that the impossibility of the quintic formula is not a failure of human ingenuity, but a fundamental mathematical truth.

Demo 15: Solvable Groups

A group is solvable if it can be built up from abelian pieces. We explore the derived series - a sequence of commutator subgroups that either reaches the trivial group (solvable) or gets stuck (not solvable). The symmetric groups Sₙ for n ≥ 5 are NOT solvable!

Solvable Groups

A group G is solvable if you can build it up from abelian pieces. More precisely, G is solvable if it has a subnormal series where each quotient is abelian.

Definition (via derived series): The derived series of G is:
⋯
where G(i+1) = [G(i), G(i)] is the commutator subgroup.
G is solvable if the derived series eventually reaches the trivial group {e}.
…
is Solvable
The derived series reaches {e}
Every abelian group is solvable. The derived series reaches {e} in one step.

Derived Series

G(0)
…
Order: 4
commutator
G(1)
…
Order: 1
✓ Reached trivial group!
Explanation: Abelian groups have trivial commutator subgroup [G,G] = {e}

Key Facts About Solvable Groups

All Abelian Groups
Every abelian group is solvable because [G,G] = {e} in one step.
Sₙ for n ≤ 4
S₁, S₂, S₃, and S₄ are all solvable. This is why polynomials of degree ≤ 4 can be solved by radicals!
Sₙ for n ≥ 5
For n ≥ 5, Sₙ is NOT solvable because it contains the simple group Aₙ. This is the heart of the unsolvable quintic!
Simple Groups
A non-abelian simple group (like A₅) cannot be solvable because its derived series gets stuck at itself.

Demo 16: Radical Extensions

A radical extension is built by repeatedly adjoining nth roots - this is precisely what we do when we "solve by radicals" using formulas like the quadratic formula. We'll see examples from the quadratic through the quartic (which work) and the quintic (which doesn't).

Radical Extensions

A radical extension is built by repeatedly adjoining nth roots. This is precisely what we do when we "solve by radicals" - we use formulas involving square roots, cube roots, etc.

Definition: An extension E/F is a radical extension if there exists a tower:
⋯
where each Ki+1 = Ki(αi) with αin ∈ Ki for some n (i.e., αi is an nth root of an element in Ki).
Solvable by Radicals
Can be expressed using radicals
Polynomial:
⋯
Galois Group:
…
✓ Solvable group
Radical Extension:
Yes
The quadratic formula x = (-b ± √Δ)/2a requires taking a square root of the discriminant. This is a radical extension of degree 2.

Radical Tower Construction

Step 0
…
Adjoin
…
(degree 2 root)
Step 1
…
Radicals Used:
…
2th root of
…

The Fundamental Connection

Solvable by Radicals ⟺ Solvable Galois Group
A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
  • Degrees 1-4: The symmetric groups S₁, S₂, S₃, S₄ are all solvable, so polynomials of degree ≤ 4 can be solved by radicals. This is why we have the quadratic, cubic, and quartic formulas!
  • Degree 5+: For n ≥ 5, Sₙ is not solvable (because Aₙ is a simple non-abelian group). Therefore, the general polynomial of degree 5 or higher cannot be solved by radicals.
  • Special cases: Some specific quintics (like x⁵ - 1) have smaller, solvable Galois groups, so they CAN be solved by radicals!

Demo 17: The Fundamental Theorem for Solvability

Galois' Great Discovery: A polynomial is solvable by radicals if and only if its Galois group is a solvable group. This theorem connects two seemingly different concepts - an algebraic property (solvability by radicals) and a group-theoretic property (group solvability) - in a profound way.

The Fundamental Theorem

Galois' Great Discovery
⋯
This theorem connects two seemingly different concepts: an algebraic property (solvability by radicals) and a group-theoretic property (solvability of the Galois group). This is the heart of Galois Theory!

Main Theorem (Galois)

⋯
This is the crowning achievement of Galois Theory! It reduces the algebraic question "can we solve this polynomial?" to the group-theoretic question "is the Galois group solvable?"

Examples: Polynomial Degrees 2-5

Polynomial:
…
Galois Group:
…
✓ Solvable group
Formula:
⋯
S₂ is abelian (cyclic), hence solvable. Quadratic formula works!

Historical Significance

For centuries, mathematicians searched for formulas to solve polynomial equations. The quadratic formula was known to ancient civilizations. In the 16th century, Italian mathematicians discovered formulas for cubics and quartics.

But the quintic resisted all attempts. Many believed a formula must exist - it was just too complicated to find. In 1824, a young Norwegian mathematician named Niels Henrik Abel proved that no such formula exists for the general quintic.

Évariste Galois (1811-1832) went further. At age 20, he developed the theory that bears his name, showing exactly why the quintic can't be solved: the symmetric group S₅ is not solvable. His work unified algebra and group theory, creating one of the most beautiful theories in mathematics.

Tragically, Galois died in a duel the night after writing down his ideas. His revolutionary work wasn't fully understood until decades after his death.

Key Takeaways

The Power of Galois Theory
Galois Theory transforms algebraic questions into group-theoretic questions. Instead of asking "can we solve this equation?" we ask "is this group solvable?"
Why No Quintic Formula
The impossibility of the quintic formula is not because we haven't been clever enough - it's a fundamental mathematical truth. S₅ is not solvable, so the general quintic cannot be solved by radicals.
Special Cases Can Work
While the general quintic has no formula, specific quintics (like x⁵ - 1) can be solved if their Galois group is smaller and solvable.
Beautiful Mathematics
This theorem is considered one of the most beautiful results in all of mathematics, connecting algebra, geometry, and group theory in a profound way.

Demo 18: The Unsolvable Quintic

The culmination: we see specific examples of quintic polynomials that cannot be solved by radicals. The general quintic has Galois group S₅, which is not solvable (because it contains the simple non-abelian group A₅). This is why there is no quintic formula!

The Unsolvable Quintic

One of the most famous results in mathematics: the general quintic polynomial cannot be solved by radicals. There is no formula like the quadratic formula for degree 5 polynomials!

No Quintic Formula Exists
You cannot solve x⁵ + ax⁴ + bx³ + cx² + dx + e = 0 using only +, -, ×, ÷, and nth roots (radicals). This is a fundamental mathematical impossibility, not a lack of cleverness!
NOT Solvable by Radicals
No radical formula exists
Polynomial:
⋯
Galois Group:
…
Order: 120
Roots:
…
The Galois group of the general quintic over ℚ(a,b,c,d,e) is S₅, which is not solvable. Therefore, there is NO formula using +, -, ×, ÷, and radicals that solves all quintics.

Why Can't We Solve the Quintic?

1. The Galois Group is S₅
The general quintic has Galois group S₅ (the symmetric group on 5 elements), which has 120 elements.
2. S₅ Contains A₅ (Simple Group)
S₅ contains A₅ (the alternating group) as a normal subgroup. A₅ is simple and non-abelian - it has no nontrivial normal subgroups.
3. S₅ is Not Solvable
The derived series of S₅ is: S₅ → A₅ → A₅ → A₅ → ... It gets stuck at A₅ and never reaches the trivial group. Therefore S₅ is not solvable.
4. Galois' Theorem
By the Fundamental Theorem of Galois Theory for solvability: a polynomial is solvable by radicals if and only if its Galois group is solvable. Since S₅ is not solvable, the general quintic is not solvable by radicals.

What CAN We Do About Quintics?

Numerical Methods
We can compute roots to arbitrary precision using Newton's method and other numerical techniques. The roots exist - we just can't write them with radicals!
Special Functions
There exist formulas for quintics using special functions (like elliptic integrals or Bring radicals), just not using ordinary radicals.
Solvable Quintics
Some specific quintics (like x⁵ - 1 or x⁵ - 2) have smaller Galois groups that ARE solvable, so these CAN be solved by radicals.
Understand the Structure
Even without explicit formulas, Galois Theory tells us about the structure of the roots and their relationships - profound information!

The Beauty of Impossibility

The unsolvability of the quintic is not a failure - it's a profound mathematical truth. Galois Theory doesn't just tell us we can't solve the quintic; it tells us why we can't, connecting the algebraic structure of polynomials to the group-theoretic structure of symmetries. This is one of the crowning achievements of 19th century mathematics and remains central to modern algebra.

The Journey Complete

You now understand one of the deepest results in all of mathematics:

Why the Quintic Formula Doesn't Exist
1.The general quintic polynomial has Galois group S₅
2.S₅ is not solvable (its derived series gets stuck at A₅)
3.By Galois' Theorem: solvable by radicals ⟺ solvable Galois group
∴The general quintic cannot be solved by radicals
✓
Solvable Groups - Groups that can be built from abelian pieces
✓
Radical Extensions - Field extensions built by adjoining nth roots
✓
The Connection - Solvable groups ⟺ solvable by radicals
✓
The Impossibility - Why there's no quintic formula
Galois' Legacy
Évariste Galois died at age 20 in 1832, the night after frantically writing down these ideas. His work was so revolutionary that it wasn't fully understood for decades. Today, Galois Theory stands as one of the crowning achievements of mathematics, unifying algebra, geometry, and group theory in a profound and beautiful way.

Next: Explore applications of Galois Theory to cyclotomic fields, geometric constructions, and real-world problems!