Galois’ Enduring Legacy
Benjamin Skuse
For Bảo Châu Ngô (Fields Medal – 2010), the Galois group underpins his 2009 seminal contribution to mathematics of proving the ‘Fundamental Lemma’ of the Langlands program, a collection of mathematical conjectures linking many areas of mathematics by 2018 Abel Prize recipient Robert Langlands. But in his half-hour lecture at the 11th Heidelberg Laureate Forum, Ngô wisely chose not to spend his time giving a primer in Galois theory or, even worse, try to outline the Langlands program and its associated Fundamental Lemma. These topics are too broad, abstract, and complicated to explain in such a short time.
Instead, Ngô provided attendees with just a taste of what the Galois group is in order to expose why Galois theory was important to progress in mathematics at the time of its inception, and why it remains central to progress today: a way of thinking that allows mathematicians to look at the underlying structure and form of mathematics.
Fierce Intelligence and Originality
Genius mathematician, staunch French Republican, and unlucky duellist Évariste Galois was only on this Earth for 20 short years, but his legacy across many branches of mathematics has endured for almost 200, and will likely continue long into the future.
Galois’ most important work came in founding what would later become known as group theory. He posited three principles for what a group is, and used these to develop more properties of groups. These properties can be used to compare groups with other groups that seem unrelated.
This technique can then be used as a means for comparing types of algebraic equations, and the solutions of these equations. More specifically, Galois groups contain all the symmetries between solutions of polynomial equations; in other words, the permutations of the roots that preserve all relations between them.
To anyone outside mathematics, this seems trivial – presenting what already exists and is known in a slightly different way, shuffling papers if you will. In reality, it was and continues to be a revelation.
From the Babylonians to Renaissance Italy
Ngô began his talk with a history lesson: “The solution for the quadratic equation that we all learn in school, you can find some equivalent form of that in Babylonian tablets that date to 2000 years BC,” he said, referring to the solution of the general form of a quadratic equation \(x^2 + bx + c\):
\[ x = \frac{-b \pm \sqrt{b^{2} – 4ac}}{2a}.\]
“It’s much more difficult when you move to cubic equations, but you can find solutions to a large number of cubic equations in Chinese and Persian [texts] dating around the 10th Century,” he added. In fact, Ngô revealed that the general forms of the solutions to cubic equations and even quartic equations were discovered before Galois’ time as well.
During the Renaissance, Italian mathematicians Gerolamo Cardano and Niccolò Tartaglia wrote down the very complicated general form for solutions of cubic equations. “Very few of us would be able to find this by ourselves,” added Ngô. “It’s elementary but it has a series of very clever and tricky changes of variables.” This breakthrough was swiftly followed by an even more complicated solution for quartic equations by another Italian mathematician Lodovico Ferrari.
Here, Ngô took a moment to pause and reflect. “But what do we mean by solution?” he asked the audience. “What we’re looking for is some kind of formula [involving] the coefficients of the polynomials and then we can use the four operations (plus, minus, multiplication, division) and take new roots – but then of course we have an ambiguity about taking roots because there are several choices.”
Abstraction Leading to Understanding
Galois theory removes this ambiguity completely. It brings together all the roots of the equation in question and describes all the symmetries between them. A symmetry between roots is where one root can be replaced by another without it affecting the answer. For example, any expression involving only adding or multiplying \(\sqrt{2}\) will have the same answer as the same expression with \(\sqrt{2}\) replaced by -\(\sqrt{2}\).
By taking a step back from the algebraic equations themselves, Galois theory revealed their underpinning structures, and Galois could very simply and eloquently tackle problems in mathematics that had only recently been resolved by complex means.
Ngô gave an example. “The Abel–Ruffini theorem showed it was impossible to find solutions to [degree 5 and higher] types of equations in general form – this was a spectacular result,” he said. “But if I give you one equation, it does not tell you whether I can solve it by radicals or not.” In other words, the theorem did not explain whether, given a particular equation, there exist solutions with rational coefficients using only rational numbers and the operations of addition, subtraction, division, multiplication, and finding nth roots.
“With the Galois group, you can reprove the Abel–Ruffini theorem, and you can use calculations with the Galois group to recover the tricky calculations of Tartaglia and Ferrari and so on,” said Ngô. Moreover, the degree 5 polynomials that are solvable are precisely those whose Galois group is solvable. In other words, Galois theory can be used to say whether a particular equation is solvable by radicals.
The Modern Era
“The whole point of Galois theory is this move from studying algebraic equations into a completely different object, some abstract group [where] the solution of the equation can be expressed in these very simple forms,” explained Ngô. As became clear much later when mathematicians started to appreciate Galois’ insights, this abstraction has been fundamental in allowing Galois theory to act as a fundamental bridge between important mathematical disciplines, and indeed other disciplines.
For instance, Galois theory introduced the abstract algebraic concept of finite fields. As it turned out, finite fields have become central to everything from defining an algorithm, to public cryptography, tomography, and building good computer networks. These fundamental, pervasive, and enduring qualities of Galois theory are why it has been described by the likes of 1994 Fields Medallist Efim Zelmanov (during his Heidelberg Lecture at the 2024 Lindau Nobel Laureate Meeting) as: “The golden standard of beauty in mathematics.”
To demonstrate how Galois theory pervades modern pure mathematics to a mixed audience is no mean feat. Ngô started with advances in topology in the 20th Century. “The torus is the first non-trivial object in topology and associated with it is the ‘fundamental group’,” he explained, where the fundamental group refers to a group associated with a topological space that records information about the basic shape, or holes, of that space.
In the torus example, if you plot a loop around the surface of the torus longitudinally and another on the meridian in the interior of the torus, there is no way to continuously deform between the two, so these are distinct. As a result, it is possible to construct a space that forms a torus just using these two types of loops. This can be expressed as the fundamental group of the torus, \( \mathbb{Z}^{2} \).
“This doesn’t seem to have much to do with Galois theory, but it does,” explained Ngô. “The ‘covering theory’.” It was Alexander Grothendieck (Fields Medal – 1966) in the 1960s who brought all this together, bridging Galois groups in number theory and fundamental groups in topology.
Though details are left to the interested reader, a covering is essentially a map between topological spaces that acts like a projection of multiple copies of a base space onto itself. So, the trivial covering space of a torus can be sketched out like a spiral staircase ending in a doughnut, the base space. Under certain restrictions and conditions, the fundamental group of a given base space is analogous to the Galois group. And from this, connections and parallels between topological spaces and fields are easily exposed, providing new insights in both subjects.
Ngô then fast-forwarded to the present day. He said that some of the biggest questions in arithmetic geometry relate to Galois theory. For instance: “How to characterize the Galois representations that can occur in cohomology [a sequence of abelian groups, usually associated with a topological space] algebraic varieties,” he asked. “We study algebraic varieties through these Galois representations, but we need to know the properties of these Galois representations.”
Although he mentioned that significant progress, including by himself, has been made in the past 20 years on this question, it is still likely to occupy mathematicians for the next 50 to 100 years. In effect, Ngô’s conclusion was that Galois’ ideas will remain relevant long after he and every member of the audience had passed away.
Fulfilling a Prophecy
On the night before the duel that took his life on 30 May 1832, Galois frantically scribbled down 60 pages of mathematical notes. These notes are often romantically credited with giving birth to group theory, even though it was his earlier work that proved decisive in this regard. However, they did contain a prophetic postscript: “Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.”
If Galois could have heard Ngô explain how his original thinking and mathematical advances continue to influence and shape mathematics in the 21st Century, no doubt he would have been satisfied that his hopes had been well and truly exceeded.
You can view Ngô’s entire lecture from the 11th Heidelberg Laureate Forum in the video below.
The post Galois’ Enduring Legacy originally appeared on the HLFF SciLogs blog.