Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois theory pops up pretty often in more advanced topics in algebraic geometry and now I want to understand why.
Thing is, I want to fulfill the following criterion:
- I want to see things geometrically from the get go. I don't want to first read about basic galois theory of fields as a purely algebraic preliminary, and see the geometric meaning only afterwards. I don't mind working hard in advance for this. Will gladly learn basic scheme theory if it will help.
- I want to work in enough generality so that the relationship between Galois groups and covering spaces is not completely shocking.
How should I learn and what should I read?
I have heard of Lenstra's notes on Galois theory for schemes, Szamuely's book on Galois groups and fundamental groups, and Borceux and Janelidze's Galois theories book, but I'm not sure where to dive in.