关于格罗滕迪克理论及相关统一理论的疑问与初学者参考资料请求

As you noted, in J.S. Milne's Fields and Galois Theory (p.105), he writes:

For Grothendieck, the classification of field extensions by Galois groups, and the classification of covering spaces by fundamental groups, are two aspects of the same theory

Hey there! Let me break down what this "same theory" is, and point you to some beginner-friendly resources to learn more.

That unifying framework is Grothendieck's generalized Galois Theory, which uses category theory and the concept of étale fundamental groups to tie these two seemingly distinct classification results together. Here's the gist:

• Classical Galois theory tells us finite Galois extensions of a field correspond to finite-index subgroups of the field's absolute Galois group.
• In topology, connected covering spaces of a space correspond to subgroups of its topological fundamental group.

Grothendieck abstracted both into a single categorical equivalence. For a connected Noetherian scheme (you can think of a field as a trivial one-point scheme, $\text{Spec}(K)$), the category of finite étale coverings of the scheme is equivalent to the category of finite sets with a continuous action of the scheme's étale fundamental group. This unifies the two cases:

• When the scheme is a field, étale coverings are exactly finite Galois extensions, and the étale fundamental group is the field's absolute Galois group.
• When the scheme is a complex algebraic variety (viewed topologically), the étale fundamental group is the profinite completion of the topological fundamental group, linking back directly to covering space theory.

Here are some great references for beginners:

• J.S. Milne's Fields and Galois Theory: You already started here, and Milne's writing is famously accessible. He also has free online lecture notes on algebraic geometry that build on classical Galois theory to introduce étale fundamental groups step by step.
• SGA 1: Revêtements étales et groupe fondamental (English translation)：This is Grothendieck's original seminar notes, the foundational source for this theory. While it's a bit dense initially, many university lecture notes simplify SGA 1's core ideas for early learners—look for notes on "étale fundamental groups" or "Grothendieck Galois theory" aimed at undergrads or first-year grad students.
• Galois Theory for Schemes: This short, focused booklet cuts through unnecessary jargon to explain the generalized Galois theory for schemes, making it ideal for someone just starting to connect classical Galois theory to algebraic geometry.
• Joe Harris's Algebraic Geometry: A First Course: If you need to build basic algebraic geometry background (like schemes, morphisms), this book is a gentle, intuitive introduction that will give you the foundation to understand étale coverings and fundamental groups.

备注：内容来源于stack exchange，提问作者JNF