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.
> 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!