**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 $L=M^H$. **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 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.