The legacy of Évariste Galois

This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.

— Hermann Weyl, Symmetry (1952)

Galois left some sixty pages of mathematics, much of it written in haste. It took half a century for them to be understood. When they were, they changed the very idea of what mathematics is: instead of studying numbers and figures, studying the transformations that leave them unchanged.

Group theory

The first book on Galois theory, the Treatise on substitutions by the Frenchman Camille Jordan, appeared in 1870. Two years later the German Felix Klein proposed, in his Erlangen Programme, that each geometry be defined by its group of transformations: Euclid’s geometry studies what does not change under rotations and translations; projective geometry, what does not change under projections. The Norwegian Sophus Lie, a fellow countryman of Abel, created the continuous groups that bear his name, which serve to study differential equations as Galois had studied algebraic ones.

The resemblance to Abel’s work is great, but there is a difference. Abel proved that no formula existed. Galois explained why, and the explanation held for all equations at once. The story of this new language, which speaks of operations instead of numbers, is also told in the chapter Galois and Cayley of this site’s book on semiotics.

Groups in nature

• Physics. In 1918 the German Emmy Noether proved that every symmetry of the laws of physics corresponds to a conserved quantity: symmetry in time gives conservation of energy, and symmetry in space, conservation of momentum. In 1961 Murray Gell-Mann organised the subatomic particles with a group called $\mathrm{SU}(3)$ and predicted a particle nobody had yet seen, the omega minus, discovered in 1964.
• Chemistry. There are exactly 230 possible symmetry groups for the arrangement of atoms in a crystal, classified in 1891. Every mineral, every metal and every crystallised protein belongs to one of them.
• Number theory. Andrew Wiles proved Fermat’s last theorem in 1995 by studying Galois groups of equations linked to elliptic curves, the same curves that grew out of Abel’s elliptic functions.
• Computing. Finite fields are in error-correcting codes and in cryptography.

The Monster

The group of 60 elements that blocks the formula for the fifth degree is simple: it cannot be cut into smaller pieces. Simple groups are the “atoms” of all finite groups, and finding all of them was one of the largest projects of twentieth-century mathematics. The classification, completed in 2004, fills tens of thousands of pages written by more than a hundred mathematicians. Besides a few infinite families, there are exactly 26 isolated simple groups. The largest of them, the Monster, has

elements, about $8\times10^{53}$.

What remained

• Theory of equations. The group of an equation and the criterion that decides which equations can be solved by radicals.
• Group theory. The word itself, the idea of a normal subgroup and the decomposition of a group into steps.
• Finite fields. The fields with $p^n$ elements, which bear his name.
• Geometry. Through Wantzel, Klein and Lie, the explanation of the impossible constructions and the idea of geometry as the study of symmetries.
• Science. Particle physics, crystallography, cryptography and codes.

To remember

• Évariste Galois lived from 1811 to 1832. He did all his work between the ages of 17 and 20.
• The group of an equation is formed by the permutations of the roots that respect every true relation between them.
• A group has closure, associativity, an identity element and inverses. The six symmetries of the triangle form the smallest non-abelian group.
• An equation can be solved by radicals exactly when its group can be cut in normal steps of prime index. For degree 5, the group of 60 elements cannot be cut.
• There is a field with $p^n$ elements for every prime $p$ and every $n$. The one with nine elements is obtained with an $i$ such that $i^2=-1$ modulo 3.
• Ruler and compass only construct numbers that need square roots. That is why the cube cannot be doubled and the $60°$ angle cannot be trisected.

Sources: Peter M. Neumann, The Mathematical Writings of Évariste Galois (2011) · Hermann Weyl, Symmetry (Princeton University Press, 1952) · Mark Ronan, Symmetry and the Monster (Oxford University Press, 2006).

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