On the history of the Artin Reciprocity Law

Question

At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians):

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define L-series for non-abelian extensions. But for them to agree with the L-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn't, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn't possibly be true...

I assume he is referring to the definition of the Artin map as sending an unramified prime ideal $\mathfrak p$ to the Frobenius element $(\mathfrak p, L/K) \in \operatorname{Gal}(L/K)$ for an abelian extension $L/K$.

Questions: Why was it so hard for other number theorists to believe it? Artin probably talked to other people around 1924-27 (he says he took 3 years to prove the theorem in the same quote later) and Chebotarev had just recently proven his density theorem. Surely the density theorem provided strong evidence of the importance of the Frobenius element.

It is also something that one can explicitly verify at least in small cases and as Artin says, one can show that it also implies the other reciprocity laws proved by then. It seems very strange to outright dismiss the idea.

Were there other candidates for what the map should be or any reason to suspect that there should be no such canonical map? Surely there must have been strong reasons for such a strong rejection.

Answer 1

As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there were no other candidates for the Artin isomorphism; reciprocity laws at the time were intimately connected to power residue symbols. Actually Euler had formulated the quadratic reciprocity law in the correct way (namely that the symbol $(\Delta/p)$ only depends on the residue class of $p$ modulo $\Delta$), but Legendre's formulation prevailed. I'm pretty sure that Hasse would not have laughed had Artin shown him right away that the general reciprocity law is equivalent to the known power reciprocity laws. Hasse did not have to wait for Chebotarev's density law to be convinced that Artin was right - by the time Chebotarev's ideas appeared it was clear that Artin must have been right. And, by the way, Chebotarev's article did a lot more than prove the importance of the Frobenius element - it provided Artin with the key idea for the proof of his reciprocity law, namely Hilbert's technique of abelian crossings.

Let me also add that in 1904, Felix Bernstein conjectured a reciprocity law in 1904 that is more or less equivalent to Artin's law in the special case of unramified abelian extensions. Its technical nature shows that it was not at all easy to guess a simple law such as Artin's from the known power reciprocity laws.