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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.

Still, I kept at it, but nothing I tried worked. Not a week went by --- for three year ! --- that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I nothing special to do, so I said, `Well, I try to prove the Reciprocity Law again.' So I went out and sat down in the garden. You see, from the very beginning I had the idea to use the cyclotomic fields, but they never worked, and now I suddenly saw that all this time I had been using them in the wrong way --- and in half an hour I had it.

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.

Still, I kept at it, but nothing I tried worked. Not a week went by --- for three years ! --- that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I had nothing special to do, so I said, `Well, I try to prove the Reciprocity Law again.' So I went out and sat down in the garden. You see, from the very beginning I had the idea to use the cyclotomic fields, but they never worked, and now I suddenly saw that all this time I had been using them in the wrong way --- and in half an hour I had it.

Addendum (2011/11/21) In Recountings (edited by Joel Segel, A K Peters Ltd, Natick, Mass.), Arthur Mattuck recounts a conversation with Emil Artin about his reciprocity law:

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.

Still, I kept at it, but nothing I tried worked. Not a week went by --- for three year ! --- that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I nothing special to do, so I said, `Well, I try to prove the Reciprocity Law again.' So I went out and sat down in the garden. You see, from the very beginning I had the idea to use the cyclotomic fields, but they never worked, and now I suddenly saw that all this time I had been using them in the wrong way --- and in half an hour I had it.

1. Here is what Tate says in his account of the General Reciprocity Law in the AMS volume on Hilbert's problems :

With this work of Takagi the theory of abelian extensions --- "class field theory" --- seemed in some sense complete, yet there was still no general reciprocity law. It remained for Artin to crown the edifice with such a theorem. He conjectured in 1923 and proved in 1927 that there is a natural isomorphism $$ > C_K/N_{L|K}C_L\buildrel\sim\over\to\operatorname{Gal}(L|K) > $$ which is characterised by the fact that...

And a little later :

How did Artin guess his reciprocity law ? He was not looking for it, not trying to solve a Hilbert problem. Neither was he, as would seem so natural to us today, seeking a canonical isomorphism, to make Takagi's theory more functorial. He was led to the law by trying to show...

Read him.

2. Here is a toy example --- not unrelated to class field theory --- of how a bijection can be more natural than others. Let $p$ be a prime number and let $K$ be finite extension of $\mathbb{Q}_p$ containing a primitive $p$-th root of $1$. There are only finitely many degree-$p$ cyclic extensions $L|K$, and there are only finitely many vectorial lines in the $\mathbb{F}_p$-space $K^\times/K^{\times p}$. In fact the two sets have the same number of elements, but the only natural bijection is $$ L\mapsto\operatorname{Ker}(K^\times/K^{\times p}\to L^\times/L^{\times p}), $$ of which the reciprocal bijections can be written $D\mapsto K(\root p\of D)$.

It follows that the number of degree-$p$ cyclic extensions $L|K$ is the same as the number of hyperplanes in $K^\times/K^{\times p}$. But is there a natural bijection between these two sets ? You will agree that $L\mapsto N_{L|K}(L^\times)/K^{\times p}$ is as natural a bijection as there can be.

1. Here is what Tate says in his account of the General Reciprocity Law in the AMS volume on Hilbert's problems :

With this work of Takagi the theory of abelian extensions --- "class field theory" --- seemed in some sense complete, yet there was still no general reciprocity law. It remained for Artin to crown the edifice with such a theorem. He conjectured in 1923 and proved in 1927 that there is a natural isomorphism $$ > C_K/N_{L|K}C_L\buildrel\sim\over\to\operatorname{Gal}(L|K) > $$ which is characterised by the fact that...

And a little later :

How did Artin guess his reciprocity law ? He was not looking for it, not trying to solve a Hilbert problem. Neither was he, as would seem so natural to us today, seeking a canonical isomorphism, to make Takagi's theory more functorial. He was led to the law by trying to show...

Read him.

2. Here is a toy example --- not unrelated to class field theory --- of how a bijection can be more natural than others. Let $p$ be a prime number and let $K$ be finite extension of $\mathbb{Q}_p$ containing a primitive $p$-th root of $1$. There are only finitely many degree-$p$ cyclic extensions $L|K$, and there are only finitely many vectorial lines in the $\mathbb{F}_p$-space $K^\times/K^{\times p}$. In fact the two sets have the same number of elements, but the only natural bijection is $$ L\mapsto\operatorname{Ker}(K^\times/K^{\times p}\to L^\times/L^{\times p}), $$ of which the reciprocal bijections can be written $D\mapsto K(\root p\of D)$.

It follows that the number of degree-$p$ cyclic extensions $L|K$ is the same as the number of hyperplanes in $K^\times/K^{\times p}$. But is there a natural bijection between these two sets ? You will agree that $L\mapsto N_{L|K}(L^\times)/K^{\times p}$ is as natural a bijection as there can be.

One last point : Given a hyperplane $H\subset K^\times/K^{\times p}$, how do you recover the degree-$p$ cyclic extension $L|K$ such that $H=N_{L|K}(L^\times)/K^{\times p}$ ? Answer : use the natural reciprocity isomorphism $K^\times/K^{\times p}\to\operatorname{Gal}(M|K)$, where $M|K$ is the maximal elementary abelian $p$-extension, to identify $H$ with a subgroup of $\operatorname{Gal}(M|K)$, and take $L=M^H$.