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A necessary condition for solvability by radicals

This handout for the University of Toronto's MAT401 course provides a Galois theory-based proof of a necessary condition for the solvability of algebraic equations by radicals, while deferring the sufficient condition to a subsequent note.

Original authors: Askold Khovanskii (Department of Mathematics, University of Toronto, Canada)

Published 2026-04-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Askold Khovanskii (Department of Mathematics, University of Toronto, Canada)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a very tricky puzzle: Can a specific algebraic equation be solved using only basic arithmetic operations (addition, subtraction, multiplication, division) and taking roots (square roots, cube roots, etc.)?

This is what mathematicians call "solving by radicals."

For centuries, people knew how to solve simple equations (like $x^2 + 2x + 1 = 0$ ). But for equations with higher powers (like $x^5 + \dots = 0$ ), it was a mystery. This paper, written by mathematician Askold Khovanskii, explains the necessary condition for solving these puzzles. In plain English, it tells us: "If you can solve an equation with radicals, its underlying 'symmetry structure' must be simple enough to be broken down into small, manageable steps."

Here is the story of the paper, broken down with everyday analogies.

1. The Players: Fields and Symmetries

To understand the proof, we need two main characters:

The Field ( $K$ ): Think of this as your toolbox. It contains the numbers you start with (like 1, 2, $\pi$ , etc.).
The Splitting Field ( $E$ ): This is the ultimate workshop. It's a place where you go to find all the solutions to your equation.
The Galois Group: This is the security guard or the symmetry manager. It keeps track of how the solutions can be swapped around without breaking the rules of the math.

The Big Idea: The paper argues that the "complexity" of the equation is directly tied to the "complexity" of this security guard (the Galois Group).

2. The Goal: Breaking the Wall

The paper asks: When can we break down a complex equation into simple steps (radicals)?

Imagine the solutions to your equation are locked behind a series of doors.

Solving by radicals means you have a master key that opens these doors one by one.
The Galois Group represents the lock mechanism. If the lock is too complicated (like a chaotic, tangled knot), no simple key will work.

3. The Strategy: Building a Ladder (The Normalization)

The paper introduces a clever trick called Normalization.

Imagine you have a messy chain of rooms (a "chain of radical extensions") where you are trying to get from the ground floor to the top floor. The problem is, some rooms are messy, and the doors don't line up perfectly. You can't easily see the whole building's structure.

Khovanskii's Solution:
Instead of looking at the messy chain, he builds a perfect, symmetrical tower (a "nested set of normal radical extensions") that contains your messy chain.

Step 1: Build a base floor that is perfectly symmetrical (like a circle of roots of unity).
Step 2: Build the next floor by adding roots, but do it in a way that keeps the whole building symmetrical.
Step 3: Repeat until you reach the top.

Why do this? Because in this perfect tower, we can clearly see the "security guards" (the Galois groups) for each floor.

4. The Clue: The Security Guards are "Commutative"

Here is the magic of the paper. When you look at the security guards for each step of this perfect tower:

Lemma 8.2 & 8.3: The guards on each floor are commutative.
- Analogy: Imagine a group of people swapping seats. If the group is "commutative," it means it doesn't matter who swaps with whom first; the final result is the same. It's a very orderly, predictable group.
- If the guards are orderly, the "lock" is simple.

5. The Conclusion: The "Solvable" Chain

The paper proves a chain reaction:

If an equation is solvable by radicals, we can build this perfect tower around it.
In this tower, every single step involves a simple, orderly group of symmetries (commutative groups).
If you stack simple, orderly groups on top of each other, the whole structure is called "Solvable."
Therefore, if the equation is solvable by radicals, its Galois group (the total lock mechanism) must be a "Solvable Group."

The "Aha!" Moment:
If you encounter an equation whose Galois group is not solvable (meaning the symmetry is too chaotic to be broken down into simple, orderly steps), then it is impossible to solve that equation using radicals. No amount of cleverness with square roots or cube roots will ever work.

Summary in a Metaphor

Think of the equation as a Russian Nesting Doll.

To solve it by radicals, you must be able to open the outer doll to reveal a smaller one, then that one to reveal a smaller one, all the way down to the center.
The Galois Group is the mechanism holding the dolls together.
This paper proves that if the mechanism is too complex (like a chaotic whirlwind of gears), the dolls are fused together and cannot be opened step-by-step.
The Necessary Condition: For the dolls to be openable (solvable), the mechanism must be a series of simple, orderly gears (a solvable group).

Why This Matters

This paper doesn't just say "some equations are hard." It gives us a precise mathematical test. If we look at the symmetry group of an equation and see it's too messy, we can immediately stop trying to find a radical formula. We know one doesn't exist. This is why we can't solve the general 5th-degree equation (quintic) with a formula like the quadratic formula—the symmetry group is too wild!

Based on the provided handout by Askold Khovanskii, here is a detailed technical summary of the paper "A necessary condition for solvability by radicals."

1. Problem Statement

The paper addresses the fundamental problem in algebraic theory: determining the conditions under which a polynomial equation $P(x) = 0$ over a field $K$ (of characteristic zero) can be solved by radicals. Specifically, the author aims to prove the necessary condition for solvability: if an equation is solvable by radicals, its Galois group must be a solvable group.

While the sufficient condition (if the Galois group is solvable, the equation is solvable by radicals) is often proven using linear algebra, this note focuses on the converse direction using Galois theory.

2. Methodology

The proof strategy relies on constructing a bridge between arbitrary "chains of radical extensions" and "nested sets of normal radical extensions" where Galois groups are well-defined. The methodology proceeds in the following logical steps:

Foundational Galois Theory: The author establishes the necessary background, including the definitions of normal extensions, Galois groups, and the correspondence between intermediate fields and subgroups. Key theorems cited include the Degree formula, the lifting of automorphisms (Theorem 1.4), and the relationship between normal subgroups and normal intermediate fields.
Orbit Polynomials: The paper utilizes the concept of orbits under the Galois group action to define minimal polynomials explicitly (Theorem 3.2), ensuring that coefficients remain in the base field.
Normalization of Radical Chains:
- A standard "chain of radical extensions" ( $K = R_0 \subset R_1 \subset \dots \subset R_n$ ) is not necessarily a normal extension, meaning its Galois group is not defined.
- To overcome this, the author introduces the concept of a nested set of normal radical extensions ( $K = E_0 \subset E_1 \subset \dots \subset E_n$ ).
- Theorem 7.2 proves that for any chain of radical extensions, one can construct a "normalization." This involves:
  1. Adjoining roots of unity (splitting field of $x^N - 1$ ) to the base field.
  2. Inductively constructing fields $E_{i+1}$ as splitting fields of polynomials derived from the minimal polynomials of the radical elements in the original chain.
Group Structure Analysis: The author analyzes the Galois groups of these normalized extensions:
- Lemma 8.2: The Galois group of a cyclotomic extension (splitting field of $x^n - 1$ ) is commutative (isomorphic to a subgroup of $U(n)$ ).
- Lemma 8.3 & 8.4: If the base field contains roots of unity, the Galois group of extensions formed by adjoining roots of $x^n - a$ (or products thereof) is commutative (isomorphic to a subgroup of a direct sum of cyclic groups).
Solvability of Groups: Using Corollary 5.4, the author demonstrates that if a group has a chain of normal subgroups where the factor groups are commutative, the group itself is solvable.

3. Key Contributions

Formalization of Normalization: The paper provides a rigorous construction (Theorem 7.2) for converting a non-normal chain of radical extensions into a nested set of normal extensions. This is crucial because it allows the application of Galois correspondence to radical solvability, which is often obscured in standard textbook treatments that assume normality without explicit construction.
Explicit Orbit-Based Minimal Polynomials: The use of orbit polynomials (Definition 3.1) to prove that the constructed fields are indeed normal extensions over the base field $K$ provides a clear algebraic mechanism for the normalization process.
Decomposition of the Galois Group: The proof decomposes the Galois group of the normalized extension into a series of commutative factor groups, directly linking the algebraic operation of taking radicals to the group-theoretic property of solvability.

4. Main Results

Theorem 9.1 (Main Theorem): If a polynomial equation $P(x) = 0$ $P (x) = 0$ over a field $K$ $K$ of characteristic zero is solvable by radicals, and $E$ $E$ is the splitting field of $P$ $P$ , then the Galois group $G(E, K)$ $G (E, K)$ is solvable.
- Proof Logic:
  1. Solvability by radicals implies the existence of a radical chain containing the splitting field $E$ .
  2. This chain is normalized to a nested set of normal extensions $K \subset E_1 \subset \dots \subset E_{n+1}$ .
  3. The Galois group $G(E_{n+1}, K)$ is shown to be solvable because it possesses a chain of subgroups with commutative factor groups (derived from Lemmas 8.2–8.4).
  4. Since $E$ is a normal intermediate field, $G(E, K)$ is a quotient (factor group) of the solvable group $G(E_{n+1}, K)$ .
  5. Quotients of solvable groups are solvable; therefore, $G(E, K)$ is solvable.

5. Significance

Completeness of Galois Theory: This note completes the theoretical framework for the solvability of polynomial equations by establishing the "only if" direction. It confirms that the algebraic structure of the Galois group is the definitive obstruction to solving equations by radicals.
Pedagogical Clarity: Written as a handout for a university course (MAT401), the paper offers a streamlined, self-contained proof that avoids the more complex machinery often found in advanced graduate texts, relying instead on standard undergraduate Galois theory results.
Handling Non-Normal Extensions: By explicitly addressing the normalization of radical chains, the paper clarifies a common point of confusion in introductory Galois theory: how to apply Galois groups to extensions that are not initially normal.

In summary, Khovanskii's note provides a rigorous, constructive proof that the solvability of a polynomial equation by radicals necessitates that its Galois group be a solvable group, bridging the gap between field extensions generated by radicals and the structural properties of their associated symmetry groups.