The Ghost of Galois: A 20-Year-Old’s Notes That Still Haunt Graduate School

On May 29, 1832, Paris. Galois sat at a desk. Papers covered it. He wrote fast. Letters ran together. Some steps were missing. He had to leave something behind. The next morning he would fight a duel. He was twenty. He did not know if he would live. He gave the papers to a friend and asked him to take them to the Academy. Then he went to the dueling ground.

The story that matters is mathematical, not sentimental. Galois asked a concrete question: can the roots of a polynomial equation be written using addition, subtraction, multiplication, division, and root extraction from the coefficients?

Quadratic equations have a formula. Cubic and quartic equations have formulas. The quintic? Lagrange looked at it. Ruffini looked at it. Abel proved the general quintic cannot be solved by radicals. But Abel did not give a way to decide whether a specific equation was solvable. Galois wanted a criterion.

He looked at the roots. An equation has n roots x1,…,xn​. They have relations. Some relations do not change when you permute the indices. Others do. Galois selected the permutations that preserve every algebraic relation with rational coefficients. These permutations form a set. He called it “le groupe.”

The group describes the symmetry of the roots. Given only rational numbers, it describes which roots cannot be told apart.

Then he adjoined. If we already know 2​, some roots can be written, and some relations become fixed. Fewer permutations are allowed. The group shrinks. He called this “adjonction.”

Then he studied how the group decomposes. If it can be split step by step, each step clear, the equation can be solved by radicals. If it cannot, the equation cannot. He called this “décomposition propre.”

The general quintic has a group of 5!=120 permutations. Inside it is a 60-element simple group, A5. It cannot be split further into cyclic groups. So the general quintic has no solution by radicals.

This is the original Galois theorem. It is not modern. It has no field extensions, no automorphisms, no normal subgroups, no quotient groups, no composition series. It has roots, permutations, adjoining, and decomposition. The ideas were enough.

After Galois died, the manuscript almost vanished. In 1843, Liouville got it. He was studying integrals. He understood it. He announced at the Academy that he had found the posthumous work of a genius. In 1846, he published it in the Journal de mathématiques pures et appliquées. The world began to read Galois.

Reading was hard. The manuscript was messy, jumpy, full of “obvious.” Leading mathematicians frowned. Camille Jordan turned it into a course.

In 1870, Jordan published Traité des substitutions et des équations algébriques. He reorganized Galois in the language of permutation groups. He defined normal subgroups, quotient groups, simple groups, and composition series. The normal subgroup made Galois’s decomposition rigorous. The quotient group described the smaller group after adjoining. The simple group was the atom. The composition series was the chain that splits a group until it cannot be split.

Jordan said: an equation is solvable by radicals if and only if every simple group in the composition series of its Galois group is cyclic. The composition series of S5​ contains A5. A5​ is a 60-element simple group. It is not cyclic. So S5​ is not solvable.

Jordan’s work was complete. But he stayed inside the permutation of roots. The next step was to leave it.

Dedekind stopped staring at roots. He stared at fields. A field is a set where you can add, subtract, multiply, and divide, and the result stays in the set. The rational numbers form a field, Q. Adjoin 2​ and you get Q(2). This is a field extension. Galois’s adjoining became field extension. His group became the automorphism group. Automorphisms fix the base field and preserve addition and multiplication. Galois theory moved from permutations of roots to symmetries of fields.

Dedekind also introduced ideals. An ideal is a subset of a ring, closed under addition and absorbing multiplication. To study algebraic objects, start with their internal substructures. This idea freed Galois theory from concrete polynomials.

In the 1920s, the Göttingen school pushed axiomatization. Emmy Noether led it. Emil Artin used linear algebra to treat field extensions as vector spaces. He gave the textbook version:

A finite extension L/K is Galois if and only if the size of Gal⁡(L/K) equals [L:K]. Subgroups of the Galois group correspond one-to-one with intermediate fields. Normal subgroups correspond to normal extensions. Quotient groups correspond to Galois groups of intermediate extensions. A polynomial is solvable by radicals if and only if its splitting field sits at the top of a radical extension tower. This is equivalent to the Galois group being solvable.

At this point, Galois theory became what graduate students see today. Group and field, two sides of one thing.

Then Grothendieck turned it into geometry. Classical Galois theory works well over C. Over Q or Fp​, the usual topology fails. Grothendieck used schemes and étale topology. He defined the étale fundamental group. When X=Spec⁡(K), this group is the absolute Galois group. When X is a smooth algebraic variety over CC, this group is the profinite completion of the topological fundamental group. Finite quotients of the étale fundamental group correspond to finite étale covers. Classical Galois theory, covering spaces, and class field theory entered one framework.

The quintic t5−t+x=0 is no longer just an equation. It defines a cover X→Ax1​. At each point of the base, the fiber is the five roots. Go around a loop, and the five roots permute. The monodromy group is S5. Solvability by radicals means the cover can be decomposed into cyclic covers. The cover controlled by S5​ cannot be decomposed this way. So the quintic has no solution by radicals.

Same conclusion. Different viewpoint.

Now the graduate student. Why is it painful?

The student does not learn Galois’s problem. The student learns the answer two hundred years later. The textbook defines groups, rings, fields, ideals, modules. Then algebraic extensions, minimal polynomials, splitting fields, normal extensions, separable extensions. Then automorphism groups, Galois extensions. Then the Galois correspondence. In one semester, the student walks the road of Jordan, Dedekind, Artin, Noether, and Grothendieck. Historical motivation is removed. Definitions arrive without a problem.

Galois worried about: Why can the quintic not be solved? What does root symmetry mean? Why does adjoining a radical shrink the group? The textbook says: let L/K be a finite Galois extension, G=Gal⁡(L/K). Then H↦LH is an order-reversing one-to-one correspondence. If you cannot follow, leave.

The student is not stupid. The teaching order reverses discovery.

Abstraction is not the enemy. Galois’s original version depended on concrete roots and rational relations. It was intuitive but limited. Dedekind and Artin rewrote it with fields and automorphisms. That rewrite let it spread into number theory, algebraic geometry, and Langlands. Without abstraction, Galois theory would be a clever trick for equations. But the abstraction was compressed, and the historical motivation was removed.

Galois theory is a large subject. Basic Galois theory is required for algebra graduate students. Modern frontiers include absolute Galois groups, inverse Galois problems, Langlands, and motivic theory. Many mathematicians spend a career in one small part.

The entrance is still Galois’s question: can the roots of an equation be expressed by radicals?

From that question, group theory, field theory, geometry, and number theory open up.

The library closing music played. The graduate student closed the book. Tomorrow is an exam on the Galois correspondence. He turned to the exercise: prove that S5​ is not solvable.