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am I correct in thinking that the frobenius density theorem (it says that the Dirichlet density of the set of primes of K that split completely in an extension L is 1/[L:k]) is sort of one of the main reasons why class field theory is a "thing" anyway? (i.e. look for congruence relations to determine the set of primes that split). This theorem tells us that an extension L of K is uniquely determined by the set of primes of K that split completely in L.

So my questions are: is this a correct way to think about the importance of this theorem? is there another way to see that an extension is uniquely determined by the splitting of primes? (is there a non-analytic way to see this fact?)

thank you

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Density theorems served as one of the main historical motivations of class field theory, but today the importance of class field theory goes far beyond that. Class field theory is ubiquitous in modern arithmetic geometry, and it would be very difficult to summarize precisely the role that it plays throughout the various branches (which overlap and interact in many subtle and surprising ways). For example, class field theory plays an important role in the study of the various kinds of $L$-functions, in Galois and étale cohomology, in the study of rational points on algebraic varieties, in the Langlands program, in Iwasawa theory...

However, it's always good to have a personal motivation in mind. It's hard to know the importance of something until we understand how it fits with the pieces around it (and I don't think we ever understand completely - I think mathematics is organic, rather than made of stone). So we have to constantly make up our own ways of thinking about things, and about their importance. Theorems such as Dirichlet's theorem on primes in arithmetic progression are beautiful and easy to state, and they served as one of the main historical inspirations for the development of the theory. So you are correct to think about it as an important aspect of the theory. And as you learn more number theory, you will discover new ways of thinking about old things, and your appreciation for them will only increase.

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I don't know a proof that a Galois extension is determined by the primes that split besides the argument with $L$-functions.

One way to think about the relationship with CFT is as follows:

  • The existence thm. of CFT says that, for each conductor $\mathfrak m$, there is an extension (the ray class field of conductor $\mathfrak m$) such that a prime splits completely in this extension iff it is trivial in the ray class gp. of conductor $\mathfrak m$.

  • Furthermore, this extension is abelian, with Galois gp. isomorphic to the ray class group.

  • Furthermore, every abelian extension is contained in a ray class field.

Class field theory emerged over a period of time, as conjectures and then as theorems (roughly between 1860 and 1920), as each of these statements was discovered, and people realized the relationship between splitting conditions given by congruences and abelian extensions.

It's good to think about cyclotomic extensions of $\mathbb Q$, which are manifestly abelian, and also easily seen to be ray class fields. And then one has the Kronecker--Weber theorem.

So ray class fields are generalizations of cyclotomic extensions --- in that they generalize the splitting properties of cyclotomic extensions. It's not so obvious they exist, though --- in general there is no direct construction, unlike the cyclotomic case. Proving that they exist, and that they are abelian with the correct Galois group, was a major theorem; but this is just the analogue of the existence of cyclotomic extensions of $\mathbb Q$. Proving that every ab. extension is contained in a ray class field is then another result again, generalizing Kronecker--Weber.

Artin discovered his map, and his reciprocity law, after CFT was first proved. One of the important aspects of the Artin map is that it gives an a priori relationship between the group of (unramified) fractional ideals and the Galois group in the case of an abelian extension.

I'm starting to digress a bit here. So let me just summarize by saying that, yes, the idea that ray class fields are determined by certain splitting properties is one of the pillars of class field theory. The other main pillar is the (completely non-obvious) relationship between such congruence based splitting properties and abelianness of extensions.

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  • $\begingroup$ Thank you for the insightful reply! It's more or less what I was looking for $\endgroup$
    – Elliot
    Jul 25, 2014 at 18:17

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