Galois Groups and the Symmetries of Polynomials
Introduction
Before being mortally wounded in a duel at age 20, Évariste Galois discovered the hidden structure of polynomial equations. By studying the relationships between their solutions — rather than the solutions on their own — he created new concepts that have since become an essential part of many branches of mathematics.
No one knows why Galois found himself on a Paris dueling ground early in the morning of May 30, 1832, but the night before, legend has it that he stayed up late finishing his last manuscripts. There he wrote:
Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.
Galois’ edict emerged from a mathematical predicament. In the 1500s, mathematicians had studied polynomials like x2 − 2 and x4 − 10x2 + 22. They had tried to find simple formulas that would allow them to calculate the roots of those polynomials — the values of x that make the equation equal zero — but could only find them when the highest exponent was no greater than 4.
Beyond that, Galois himself proved that no such formulas exist. So he devised a new way of studying roots: Instead of calculating them exactly, he realized he could study the algebraic relationships between them — focusing on their complexities, rather than their appearances.
In spirit, his perspective was similar to considering a shape’s different symmetries. These are the various ways of reorienting the shape so that it still looks the same, such as rotating a square by 180 degrees. Symmetries between a polynomial’s roots are ways of swapping them so that they maintain the same algebraic relationship.
And just as some shapes have more symmetries than others (a circle has infinitely many; a square has just eight), you can rearrange the roots of some polynomial equations more freely than you can rearrange the roots of others.
“Some ways of rearranging the roots can be incompatible with the rules of algebra. In this sense, the roots might not be entirely interchangeable with one another,” said Brian Conrad of Stanford University.
The extent to which roots can be swapped with each other while maintaining algebraic consistency is a subtle property that tells mathematicians a lot about how to recognize features of polynomials that can’t be seen just by looking at them. It’s easiest to see with examples. Let’s take a look at two, each of which has three roots (since the highest exponent of each is 3):
f(x) = x3 − 7x + 5
g(x) = x3 − 7x + 7
On paper, they’re nearly identical. But behind the scenes, the roots of one can be rearranged in more ways than the roots of the other.
Let’s focus on f(x) first. Here, we have three roots: a, b and c. We can combine them algebraically to make a new value by taking the product of pairs of roots and adding them together. For all cubic polynomials — those with 3 as their highest exponent — with a coefficient of 1 for the cubed term, it’s known that this particular algebraic expression made from the roots always equals the coefficient of the linear term, or the term that is raised to the first power. In our example, this is −7.
We get this algebraic equation:
ab + ac + bc = −7.
Now let’s rearrange the roots, leaving c alone, but switching a and b. We get:
ba + bc + ac = −7.
Rearranging the roots in this way preserves the algebraic relationship between them: The equation is still true because multiplication and addition are commutative, meaning that swapping the order of things — like shuffling the roots around — doesn’t change the answer. In fact, for this example, all six possible ways of rearranging the roots (including the one where they don’t change) preserves the relationship:
a, b, c: ab + ac + bc = −7
b, a, c: ba + bc + ac = −7
c, b, a: cb + ca + ba = −7
a, c, b: ac + ab + cb = −7
b, c, a: bc + ba + ca = −7
c, a, b: ca + cb + ab = −7
Now let’s look at the second polynomial, g(x) = x3 − 7x + 7. If we call the roots r, s and t, then an analogous equation to the one for f(x) holds as well:
rs + rt + st = −7.
This will be true for any cubic polynomial whose initial term is x3 and whose linear term is −7x. And again, all six of the possible arrangements still equal −7. But curiously, for g(x), not all of them are considered symmetries of the polynomial.
This is because the algebraic relationships among its roots are more complex: There is an additional special algebraic relationship that its roots satisfy. The special relationship is (r − t)(r − s)(t − s) = 7 (when you assume r is less than s, and s is less than t). Only three of the six possible rearrangements of its roots preserve both algebraic relationships: rs + rt + st = 7 and (r − t)(r − s)(t − s) = 7:
r, s, t: (r − t)(r − s)(t − s) = 7
s, r, t: (s − t)(s − r)(t − r) = −7
t, s, r: (t − r)(t − s)(r − s) = −7
r, t, s: (r − s)(r − t)(s − t) = −7
s, t, r: (s − r)(s − t)(r − t) = 7
t, r, s: (t − s)(t − r)(s − r) = 7
The three rearrangements in bold preserve all algebraic relationships among the roots, even beyond these two. Consequently, these three rearrangements are considered to be the symmetries of the polynomial.
It’s not obvious at a glance that the two polynomials have different levels of complexity, but it becomes visible when you adopt the perspective Galois invented.
Galois packaged his way of thinking in new objects — which came to be called Galois groups — that encode the complexity of the algebraic relationships between the roots of a given polynomial. Within these relationships, rearrangements of roots can be applied one after another, but they can be undone to get back to where you started — just as you can apply the symmetries of a square and then undo them to get back to the exact position you began with.
This idea reflects the general concept of a group in mathematics, which is a collection of symmetries, whether they apply to a square or the roots of a polynomial. Galois groups were the first instances of the concept of a group, and Galois’ ideas blossomed into what today is a powerful, ubiquitous area of research called group theory.
Galois groups provide a powerful perspective from which to study polynomial equations. If you know the Galois group of a polynomial, then the behavior of its roots can be understood by accessing many of the tools of group theory. The insights you’ll gain through this approach are far more illuminating than the ones you can get by performing algebra on the polynomial itself.
“[With Galois groups] you get this one piece of information, and it spreads and tells you so much more,” said David Harbater of the University of Pennsylvania.
For instance, the Galois group immediately tells you whether a polynomial can be solved at all, and it allows you to compare the underlying structure of different polynomials. Galois groups can also be used to study various mathematical objects in algebra and number theory in ways that open up solutions to problems that aren’t otherwise available.
“Turning a question about polynomials into a question about groups opens up the door to many other mathematical operations and techniques that cannot be readily described in the original language of polynomials,” said Conrad.
This expansiveness has allowed Galois groups to play a central role in many of the most celebrated mathematical projects over the last century or so. They featured in Gerd Faltings’ 1983 proof of the Mordell conjecture and Andrew Wiles’ 1994 proof of Fermat’s Last Theorem.
Galois groups are also at the heart of some of the most exciting ongoing work in mathematics today. As Quanta explained in a recent feature story, they are the linchpin of the sprawling Langlands program, which turns a question about polynomials into a more sophisticated and revealing question about the relationship between Galois groups and another special class of groups.
Though Évariste Galois’ life was cut short, his greatest achievement will continue to advance mathematics for centuries to come — though it’s hard to predict exactly how.
“[Galois groups] just have a way of appearing in surprising places,” said Jose Rodriguez of the University of Wisconsin, Madison.
