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I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about global class field theory from the corresponding local facts. This often means that global class field theory is given the idelic formulation, as local fields have already been covered.

Alternatively, I'm about to take a course on class field theory (which is the sequel to an undergraduate course on algebraic number theory and basic zeta/L-functions) which dives directly into global class field theory and will follow the original (1920s) formulations (ideal-theoretic) and proofs of the basic results.

I'm wondering what are people's opinions of the two different approaches to class field theory. Does it make more sense to start local and go global, or is it a better idea to learn the subject more historically? I asked my professor here at Princeton about it, since I was aware that Harvard Harvard's CFT course starts with local, he responded that since what we're really interested in are number fields anyway, it's much more relevant to proceed immediately with global class field theory. Thoughts?

EDIT/UPDATE: Based on input from this thread and more experience, here is the approach I've decided to follow:

1. Learn global class field theory using more elementary proofs, following something like Janusz (or another source if you don't like Janusz's style)

2. Learn the cohomology-heavy proofs of local class field theory. I particularly like Milne's notes for this.

3. Continue and learn the proof of global class field theory using cohomology of ideles. You could just continue in Milne, or try the chapters in Cassels-Frohlich

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I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about global class field theory from the corresponding local facts. This often means that global class field theory is given the idelic formulation, as local fields have already been covered.

Alternatively, I'm about to take a course on class field theory (which is the sequel to an undergraduate course on algebraic number theory and basic zeta/L-functions) which dives directly into global class field theory and will follow the original (1920s) formulations (ideal-theoretic) and proofs of the basic results.

I'm wondering what are people's opinions of the two different approaches to class field theory. Does it make more sense to start local and go global, or is it a better idea to learn the subject more historically? I asked my professor about it, since I was aware that Harvard starts with local, he responded that since what we're really interested in are number fields anyway, it's much more relevant to proceed immediately with global class field theory. Thoughts?

EDIT

EDIT/UPDATE: Based on input from this thread and more experience, here is the approach I've decided to follow:

1. Learn global class field theory using more elementary proofs, following something like Janusz (or another source if you don't like Janusz's style)

2. Learn the cohomology-heavy proofs of local class field theory. I particularly like Milne's Milne's notes for this.

3. Continue and learn the proof of global class field theory using cohomology of ideles. You could just continue in Milne, or try the chapters in Cassels-Frohlich

3 added 539 characters in body

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about global class field theory from the corresponding local facts. This often means that global class field theory is given the idelic formulation, as local fields have already been covered.

Alternatively, I'm about to take a course on class field theory (which is the sequel to an undergraduate course on algebraic number theory and basic zeta/L-functions) which dives directly into global class field theory and will follow the original (1920s) formulations (ideal-theoretic) and proofs of the basic results.

I'm wondering what are people's opinions of the two different approaches to class field theory. Does it make more sense to start local and go global, or is it a better idea to learn the subject more historically? I asked my professor about it, since I was aware that Harvard starts with local, he responded that since what we're really interested in are number fields anyway, it's much more relevant to proceed immediately with global class field theory. Thoughts?

EDIT: Based on input from this thread and more experience, here is the approach I've decided to follow:

1. Learn global class field theory using more elementary proofs, following something like Janusz (or another source if you don't like Janusz's style)

2. Learn the cohomology-heavy proofs of local class field theory. I particularly like Milne's notes for this.

3. Continue and learn the proof of global class field theory using cohomology of ideles. You could just continue in Milne, or try the chapters in Cassels-Frohlich