The Langlands Program has been progressing for a long time, with many of the big names in mathematics involved. Dr. Matthias Strauch, an Associate Professor in the Indiana University Mathematics Department, and I discussed some of the history of the field.
The story begins with linear equations, although the modern scope of research has flown far beyond. These are equations containing no squares or higher powers, no roots, no dividing by anything weird. (Dividing by numbers is fine, but not dividing by a variable.) These equations might be familiar, depending on how long it’s been since you took algebra.
Believe it or not, we can solve almost all of these. Only one thing can go wrong, and that is if, after grouping like terms, the coefficient on the x term is 0. Then solutions might not even exist. But when the coefficient is not zero, we get a solution. In fact, it has a formula.
Say we have a quadratic equation like , where a is nonzero. Now we have a more complicated formula:
But what if gets involved? Is there a formula then?
Although you might not have ever seen it, the answer is yes. If we want to solve , we use:
As you can see, this is complicated!
Is there a formula for equations with ? And to take a step back–what exactly do we mean by formula?
In order to ask if a formula exists, we have to think about what a formula really is–what you’re allowed to do. According to Dr. Strauch, we are allowed the standard algebraic operations of adding, subtracting, multiplying, and dividing, and we can take nth roots. This means we can take square roots, as in the quadratic equation, or cubic roots like in the cubic equation.
But the short answer is yes, we can find a formula for the quartic equations–those involving an term. It is much longer than the last formula.
It took mathematicians a very long time to find all these formulas, and they wondered whether more formulas existed at all. Maybe somehow there was no possible way to find an explicit formula for higher order equations which captured all the solutions by combining the coefficients in algebraic ways. Not just hard to find, but impossible.
And after some brilliant work, they found an answer. It is impossible to find a general formula for any degree higher than 4. The quartic formula is the last one.
But how did this come about? It involved many mathematicians, particularly Niels Heinrik Abel of Norway, Paulo Ruffini of Italy, and Evariste Galois, from France. As Dr. Strauch commented, “There are these points where, somehow, now we enter a new direction.”
Galois’s big idea was to look at rearrangements of the roots. Specifically, certain “admissible” rearrangements: “the idea is that any algebraic relation among the roots should hold after we permute them,” Strauch said: If two roots have a sum of 5, then after rearranging, the new roots in these roles must also have a sum of 5, and so on.
How exactly this proves the impossibility of the quintic formula is complicated, and we’ll have to let that part of the story go unfinished.
The narrative continues to change, “At some point you see that the problem you started with — it should not be the center, it is actually something … a little bit aside,” Strauch said. Mathematicians developed techniques and thought about ideas while solving this problem, and these ideas became interesting in and of themselves. So we have to take a different perspective about the theory.
“It’s not really about the solutions” to polynomials, he said. The field has advanced beyond this, and “from an advanced point of view the [results about solutions] are just corollaries or results, natural outflows” from the general theory.
This is not unheard of within mathematics. As a field develops and we try to prove some difficult theorem, we create techniques and craft ideas which we want to study in their own right, which may dwarf or subsume the old ones.
Cars look small from planes, and I bet planes look small from a space shuttle.
Until next time!