Why There Is No Quintic Formula

Every quadratic $ax^2 + bx + c = 0$ is solved by

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Note what kind of object that is: the coefficients, combined using addition, subtraction, multiplication, division and a square root. Nothing else. Feed in any $a$, $b$, $c$ and the roots come out.

The obvious question is whether every degree has one. For a long time the answer looked like yes-with-effort: cubics and quartics fell, so quintics presumably awaited a sufficiently determined algebraist.

Key idea: The answer is no, and "no" here does not mean undiscovered. It means provably non-existent, and the proof works by turning a question about formulas into a question about a group of permutations. This lesson is about how that translation works.

Babylonian scribes were completing the square by around 1650 BC. The cubic and quartic fell within a few years of each other in Renaissance Italy, both published by Cardano in 1545, the quartic being his student Ferrari's work. Then the sequence stopped for 275 years, not for want of attention: it was among the most famous open problems in mathematics.

The claim has to be stated carefully, because a careless version of it is false and a lot of people believe the careless version.

Definition: A polynomial is solvable by radicals when its roots can be written using its coefficients, the four arithmetic operations, and $n$th roots, in finitely many steps.

What is impossible is a single formula of that kind covering every quintic. Three things remain entirely possible and are worth separating:

• Quintics have five roots, guaranteed by the fundamental theorem of algebra.
• Those roots can be computed numerically to any precision you like, quickly.
• Particular quintics are often solvable by radicals, as the next-but-one step shows.

The impossibility is about a specific kind of expression, not about existence and not about computation.

In 1770 Lagrange examined every known solution method and asked what they had in common. His answer was that each one builds an auxiliary expression in the roots which takes surprisingly few distinct values when the roots are permuted, and that expression is what makes the problem collapse to a lower degree.

The symmetric functions of the roots are the coefficients themselves, and they do not move at all under permutation:

$r_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n}, \qquad r_1 r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}$

Since the coefficients are all you are given, and they are exactly what is invariant under permuting roots, any formula you build from them starts out unable to distinguish the roots at all. Solving means progressively breaking that symmetry, and each radical you take is one step of breaking.

Key idea: Permutations of the roots became the object of study. The question stopped being "what formula solves this" and became "what does this group of permutations allow".

Galois made Lagrange's observation precise. Given a polynomial, consider the permutations of its roots that preserve every polynomial relation holding among them with rational coefficients. Those permutations form a group: the Galois group of the polynomial.

It measures how indistinguishable the roots are from the coefficients' point of view. Two examples make the idea concrete:

• $x^2 - 2$ has roots $\pm\sqrt{2}$. Nothing expressible in rationals separates them, so swapping them preserves everything. The Galois group has order 2.
• $x^2 - 4$ has roots $2$ and $-2$, both rational. Any swap breaks the relation $r_1 = 2$. The Galois group is trivial.

For a general polynomial of degree $n$ with no special relations among its roots, nothing constrains the permutations at all, and the Galois group is the full symmetric group $S_n$.

Definition: The Galois group is the symmetry group of a polynomial's roots. A smaller group means the roots are more pinned down by the coefficients, and a larger one means they are more interchangeable.

Here is the correspondence that does the work. Adjoining an $n$th root to your number system cuts the Galois group down by a step, and the step it takes is always of a particular restricted kind: it replaces the group $G$ by a normal subgroup $H$ whose quotient $G/H$ is abelian.

So a formula in radicals, built from finitely many root extractions, corresponds exactly to a chain

$G = G_0 \triangleright G_1 \triangleright \cdots \triangleright G_k = \{e\}$

where each $G_{i+1}$ is normal in $G_i$ and each quotient $G_i / G_{i+1}$ is abelian. A group admitting such a chain is called solvable, and the name is not a metaphor.

Key idea: The polynomial is solvable by radicals if and only if its Galois group is solvable. That single sentence converts a three-century-old question about formulas into a finite, checkable question about the structure of a group.

Now check the symmetric groups. The general degree-$n$ polynomial has Galois group $S_n$, so everything hinges on whether $S_n$ is solvable.

$S_2$, $S_3$ and $S_4$ are all solvable, and their chains are exactly the classical solution methods: the resolvent cubic used to solve the quartic is the quotient step in $S_4$'s chain. The formulas were the group structure all along, written in the notation of the time.

$S_5$ is not solvable, and the reason is specific. It contains $A_5$, the alternating group of order 60, which is simple: it has no normal subgroup other than itself and the identity. A chain cannot get past it, and $A_5$ is not abelian, so the chain fails there and no radical formula exists.

The barrier is not size. It is that 60 elements happen to form a non-abelian simple group, and 24 do not.

Galois's result is sharper than Abel's for exactly this reason. Abel proved that no general formula exists. Galois gave a criterion that decides any specific polynomial, by computing its Galois group and asking whether that group is solvable.

Gotcha: "The quintic is unsolvable" is a compression of "the general quintic is not solvable by radicals". Taken literally it is false three times over, since quintics have roots, those roots are computable, and many individual quintics have radical solutions.

Paolo Ruffini published an argument in 1799 running to over 500 pages. It was essentially correct and contained a gap he did not notice, and it was largely ignored.

Niels Henrik Abel proved the impossibility properly in 1824, at 21, and paid to print it himself as a six-page pamphlet, compressing it so far to save money that it became hard to follow. He died of tuberculosis at 26.

Evariste Galois built the general theory as a teenager. He was rejected twice from the Ecole Polytechnique, his submissions to the Academy were lost or returned, and on 30 May 1832 he was shot in a duel and died the next day, aged 20. The night before, he wrote out his ideas in a letter to a friend, with marginal notes asking for the work to be judged for its importance rather than its completeness. It took another decade before Liouville published it and the mathematical community understood what it contained.

Key idea: Group theory did not exist and then get applied to polynomials. It was largely invented in order to answer this question, and much of the vocabulary in the previous three lessons, normal subgroups, quotients, solvability, was created for this proof.

Galois theory generalised into a dictionary: subfields of a field extension correspond exactly to subgroups of its Galois group, with the correspondence reversing inclusion. Questions about numbers become questions about finite groups, which are far more tractable.

The same machinery settled several problems that had been open since antiquity, all by the same move of showing a required field extension has the wrong degree:

In practice: The transferable lesson is the shape of the argument. When a problem resists direct attack, find a structure that every solution would have to respect, then show that structure cannot exist. That move recurs in the halting problem, in the impossibility results of distributed systems, and in lower bounds throughout computer science. Galois is where it starts.