It's all abstract nonsense until it starts having practical results lol :)
The main motivation for Galois theory was proving the insolvability of the quintic. For those not aware, there is a general formula solving the quadratic equation (i.e. solving ax^2 + bx + c = 0). That formula has been known for millenia. With effort, mathematicians found a formula for the cubic (i.e. solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order 4 polynomials). But no one was able to come up with a closed-form solution to the quintic. Galois and Abel eventually proved that a quntic formula DOES NOT EXIST. At least, it cannot be expressed in terms of addition, subtraction, multiplication, division, exponents and roots. You can even identify specific equations that have roots that cannot be expressed in those forms, for example x^5 - x + 1.
I took an entire course on it that went through the proof. It's actually very interesting. It sets up this deep correspondence between groups and fields, to the point that any theory about groups can be translated into a theory about fields and vice versa. And it provides this extremely powerful set of tools for analyzing symmetries. The actual proof is actually really anticlimactic. You have all these deep proofs about the structures of roots of polynomial equations, and at the very end you just see that the structures of symmetries of certain polynomials of degree 5 (like x^5 - x + 1) don't follow the same symmetries that the elementary mathematical operations have. Literally, the field of solutions to that polynomial doesn't map to a solvable group: https://en.wikipedia.org/wiki/Solvable_group
In almost the same breadth, it also proves that it is impossible to trisect an angle using just a compass and straightedge, a problem that had been puzzling mathematicians for millennia. It's actually almost disappointing: we spent then entire course just defining groups and fields and field extensions and all this other "abstract nonsense". And once all of those definitions are out of the way, the proof of the insolvability of the quintic takes 10 minutes, same for the proof of impossibility of trisecting an angle.
Did the solution taking 10 minutes make it seem like it was all just semantics from old faulty definitions?
How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric? Impossible things are weird conversation subjects.
> Did the solution taking 10 minutes make it seem like it was all just semantics from old faulty definitions?
I recall that I, and the rest of the class, were very suspicious of the proof. The proof took maybe 10 minutes, but it probably took another 10-15 minutes for the professor to convince us there wasn't a logical error in the given proof. Though the situation was kind of the opposite of what you would thing. We understood field extensions and the symmetries of the roots of polynomials really well. What took convincing was that any formula using addition, subtraction, multiplication, division, exponents, and rational roots would always give you a field extension that mapped to a "solvable group". The proof is essentially:
1. Any field extension of a number constructed using those mathematical operations must map to a solvable group.
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
3. There exist groups that are not solvable.
4. Therefore, there are polynomials with roots that can't be constructed from the elementary mathematical operations.
Basically the entire course is dedicated to laying out part 3, and the part we were suspicious about was part 1.
The one thing that is interesting about the proof is that it is actually partially constructive. Because there is no general quintic formula, but there are some quintics that are solvable. For instance, x^5 - 1 clearly has root x=1. And Galois theory allows you to tell the difference between those that are solvable and those that are not. It allows you to take any polynomial and calculate the group of symmetries of those roots. If that group is solvable, then all of the roots can be defined in terms of elementary operations. If not, at least one of the roots cannot.
> How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric?
So the trisection proof I don't remember as well, but looking it up it isn't very geometric. It essentially proves that trisecting an angle with a compass and straight edge is equivalent to solving certain polynomial equations with certain operations, and goes into algebra.
That said, Galois theory itself feels very "geometric" in the roughest sense of the term. Fundamentally, it's about classifying the symmetries of an object.
The reason that the proof isn't geometric, is that the algebriac proof is a proof that Euclidean geometry is incomplete. How can you use a language (any given language!) to express the idea that the selfsame language is incapable of expressing a certain concept?
You can draw a picture of trisecting an angle using an ruler (with cube-root markings) or an Archimedian sprial, which are clearly more powerful than purely Euclidean geometery, but how can you draw a picture of it being impossible without something like this?
How do you draw a picture of something that doesn't exist?
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
Just a nit, but when talking about extensions of Q, this is called the Inverse Galois Problem and it is still an open problem.
That said, you don’t actually need this strong of a statement to show general insolvability of the quintic. Rather you just need to exhibit a single extension of Q with non-solvable Galois group. I believe adjoining the roots of something like x^5+x+2 suffices.
Your story makes me picture Geometric Algebra, defining Complex Numbers or Quaternions and multiplication on them, and their symmetries and elegant combinations. Ty for sharing your math memories. Makes me wonder if Galois theory can determine valid or invalid imaginary number combinations / systems.
It solves a problem that was puzzlimg mathematicians for millennia - I wouldn't call that a footnote. You may not consider pure math applications interesting, but that doesn't make them unimportant.
Galois theory is big today because it provides a connection between a ring theory and field theory. This has huge applications for other branches of modern math - for example, number theory. But that always culminates in a pure math application, which makes it a footnote, I guess.
The main motivation for Galois theory was proving the insolvability of the quintic. For those not aware, there is a general formula solving the quadratic equation (i.e. solving ax^2 + bx + c = 0). That formula has been known for millenia. With effort, mathematicians found a formula for the cubic (i.e. solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order 4 polynomials). But no one was able to come up with a closed-form solution to the quintic. Galois and Abel eventually proved that a quntic formula DOES NOT EXIST. At least, it cannot be expressed in terms of addition, subtraction, multiplication, division, exponents and roots. You can even identify specific equations that have roots that cannot be expressed in those forms, for example x^5 - x + 1.
I took an entire course on it that went through the proof. It's actually very interesting. It sets up this deep correspondence between groups and fields, to the point that any theory about groups can be translated into a theory about fields and vice versa. And it provides this extremely powerful set of tools for analyzing symmetries. The actual proof is actually really anticlimactic. You have all these deep proofs about the structures of roots of polynomial equations, and at the very end you just see that the structures of symmetries of certain polynomials of degree 5 (like x^5 - x + 1) don't follow the same symmetries that the elementary mathematical operations have. Literally, the field of solutions to that polynomial doesn't map to a solvable group: https://en.wikipedia.org/wiki/Solvable_group
In almost the same breadth, it also proves that it is impossible to trisect an angle using just a compass and straightedge, a problem that had been puzzling mathematicians for millennia. It's actually almost disappointing: we spent then entire course just defining groups and fields and field extensions and all this other "abstract nonsense". And once all of those definitions are out of the way, the proof of the insolvability of the quintic takes 10 minutes, same for the proof of impossibility of trisecting an angle.