# Reference for Learning Global Class Field Theory Using the Original Analytic Proofs?

Hi Everyone!

I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local class field theory first or uses ideles/cohomology to prove global class field theory. This is not how it was historically done - the ideal-theoretical formulation of class field theory was proven first, using more elementary analytic methods. So I'm wondering if anyone knows of any resources which would teach these proofs.

I'm currently about to take a class which follows global class field theory in this way, and our teacher says he does not know of any textbook for this, so I'm wondering if anyone here would know.

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## 6 Answers

As far as textbooks your best bet is Janusz's "Algebraic Number Fields."

Also I tried to collect a lot of this stuff in my senior thesis. The list of references there should also be very useful. For example, I use Hecke's original approach to abelian L-functions instead of Tate's thesis which I learned from the last section of Neukirch's big book (which is otherwise a very modern book), there's Hilbert's original proof of lifting of the Frobenius from his Zahlbericht (which appears in translation and I highly recommend), and a proof of Kronecker-Weber following the original approach appears both in Mollin's "Algebraic Number Theory" and in Hilbert's Zahlbericht.

As an added bonus for non-German readers I translated (caveat, I didn't know any German at the time and relied heavily on dictionaries) Artin's beautiful paper on L-functions in the appendix which is one of the key original sources here.

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Since you mention Hilbert's proof of lifting of Frobenius, I will point out that Keith Conrad has done a very nice job translating and modernizing Frobenius's original proof: math.uconn.edu/~kconrad/blurbs/gradnumthy/frobeniuspf.pdf – David Speyer Dec 9 '09 at 16:41
It's really remarkable the way that proof (which is short, clear, and elementary) managed to disappear from the literature. Hilbert's proof is roughly along the same lines as the one in Conrad's notes, but is slightly different. – Noah Snyder Dec 9 '09 at 17:07

The nicest source that I know is Hasse's Zahlbericht (originally published as Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper in 1930 and republished in 1965), the first part of which gives the ideal-theoretic approach with Takagi's proofs, and the second part of which gives Artin's reciprocity law along with many charming explicit reciprocity laws. The original papers of Takagi (available in his collected works) are also readable. Herbrand wrote a nice summary of the state of the art in 1935 (Le développement moderne de la théorie des corps algébriques; corps de classes et lois de reciprocite (Mem. Sci. Math. 75) Paris: Gauthier-Villars. 72 p. (1935)), which is available in most university libraries.

Lang in his Algebraic Number Theory (originally based on lectures of E. Artin), in fact, follows the original proofs fairly closely, although he uses ideles and the Herbrand quotient to streamline the proofs. One can get a sense for the sort of simplification that ideles afford by reading Franz Lemmermeyer's article on the development of genus theory (in The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae Springer (2007) or at Lemmermeyer's own web page). In that article Lemmermeyer discusses the history of the famous first and second inequalities from their original appearance (with algebraic proofs that, in a sense, reappeared in Chevalley's treatment of classfield theory) in Gauss's Disquisitiones through their 19th-century Dirichlet-style analytic treatment and to the form used in the classfield theory of the 1920's. Since Lemmermeyer (following the historical sources) works with full rings of integers (without any idelic techniques), there are many tricky local terms in the formulas. Perhaps that is what you're hoping to see!

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I found Takagi's papers, but unfortunately they are in German. Does anyone know if there exists a translation? – David Corwin Dec 11 '09 at 16:19

"The" classical approach to class field theory can be found in Hasse's Marburg lectures from the early 1930s (in German, as of now). The key arguments are contained in Artin's three lectures on class field theory from 1932, given (in English) in the appendix of Harvey Cohn's book "A classical invitation of algebraic numbers and class fields".

Another nice introduction along the classical lines is provided by Iyanaga's "Class-field theory notes". Perhaps this is the book you're looking for. It's definitely more classical than Janusz. Recently, Nancy Childress has published a book on class field theory, which is at least "semi-classical".

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This isn't exactly an answer to your question, but Lenstra and Stevenhagen's article Cebatarov and his Density Theorem does a very good job covering some of the problems that lead up to Class Field Theory, including Cebatarov's original proof of his theorem.

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You might look at Harvey Cohn's book "Introduction to the Construction of Class Fields".

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You might also want to look at Narkiewicz, Elementary and analytic theory of algebraic numbers.

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