"Let’s imagine Évariste Galois survived that fateful duel and continued his work. What would he have done next? If Galois had lived past 21, he would have likely expanded and deepened his revolutionary ideas. By that time, he had already laid down the foundation of what we now call Galois theory, but there were still many directions he could have explored further.

Generalization of Galois Theory: Galois had just scratched the surface of the powerful connection between field theory and group theory. His next logical step might have been to generalize his ideas to more complex situations, such as understanding the symmetries of solutions to equations in multiple variables or extending his work to non-polynomial equations.

Exploration of Solvable Groups: Galois introduced the concept of solvable groups to determine whether a polynomial equation could be solved by radicals (using addition, multiplication, and roots). If he had lived longer, he would likely have delved deeper into classifying different types of groups and exploring their properties. This could have led to a more detailed theory of group structures, potentially anticipating parts of what we now know as modern group theory.

Linking Galois Theory with Other Areas: Galois might have explored connections between his theory and other areas of mathematics, such as geometry, number theory, or even early versions of what we now call topology. For instance, he could have investigated how the symmetries of geometric shapes relate to the symmetries of algebraic equations.

Developing Further Applications: Beyond theory, Galois could have sought applications of his work in more practical areas. Perhaps he would have applied his ideas to solving problems in physics, mechanics, or other emerging fields, where symmetry plays a crucial role.

Inspiring the Next Generation: If Galois had lived, he might have become a prominent teacher and mentor, influencing a new generation of mathematicians. His radical ideas would have challenged and inspired others to push the boundaries of mathematics even further. Galois could have spearheaded a new movement in algebra, similar to how Einstein revolutionized physics.

Refining His Work: Galois' original manuscripts were hastily written under dire circumstances. Given more time, he would likely have polished and clarified his ideas, making them more accessible to other mathematicians of the time. This might have sped up the acceptance and expansion of his ideas in the mathematical community."

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I'd give it a C for effort, I guess. I don't know enough about the topic to ask it for more specific predictions. At the end of the day it can only work with what it was trained with, so any genuine insights it comes up with will be due more to accident than inspiration.

> solutions to equations of multiple variables

Multivariate Galois theory is a thing. See e.g. https://icerm.brown.edu/materials/Slides/htw-20-mgge/Galois%...

> extending his work to non-polynomial equations

This is like a forester extending their work to non-forests. The person can learn to do other things, but those things aren't in any way an extension of forestry.

> Exploration of Solvable Groups […] Linking Galois Theory with Other Areas

This doesn't say anything.

> Perhaps he would have applied his ideas to solving problems in physics, mechanics, or other emerging fields, where symmetry plays a crucial role.

Still isn't saying anything, but if I pretend this has meaning: he was born about a century early for that.

> he might have become a prominent teacher and mentor, influencing a new generation of mathematicians.

He's far more likely to have been a political revolutionary. By the time of his death, academia had excluded him about as much as was possible.

> Given more time, he would likely have polished and clarified his ideas, making them more accessible to other mathematicians of the time.

Probably!