Artin reciprocity theorem for Hilbert class field

Question

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension)

Artin's reciprocity theorem: If $L$ is the Hilbert class field of a number field $K$, then the Artin map

$$ \left ( \frac { L/K}{ \cdot} \right) : J_K \to \text{Gal}(L/K)$$ is surjective, and its kernel is the subgroup $P_K$ of principal fractional ideals. Thus, the Artin map induces an isomorphism:

$$ Cl_K = J_K / P_K \cong \text{Gal}(L/K)$$

Question: Why this theorem is called reciprocity ? To be more precisy, my question is what this theorem actually says, and why is this a reciprocity law i.e how is this connected to the classical quadratic reciprocity law that we know from elementary number theory, and why Artin's law is a generalization of this.

Thank you in advance.

Answer 1

The statement which you refer to as Artin Reciprocity, is actually a consequence of Artin Reciprocity. The actual Artin Reciprocity Theorem is the one found on Theorem 8.2 of Cox's book. Using this result, Cox proves the classical quadratic reciprocity in Theorem 8.12. Then in Theorem 8.14 he states strong reciprocity, and this is the statement which shows the power of the Artin Reciprocity map and why the theorem bears the name reciprocity. Cox gives a short excerpt on cubic reciprocity after 8.12 and you can get $n$-th reciprocity through local computations of the Hilbert Symbol.

Answer 2

I would also recommend chapter XI, "History of class field theory", in Cassels-Fröhlich's book "Algebraic Number Theory", Academic Press, 1967, where Hasse relates Hilbert's view of class field theory as a "theory of relatively abelian fields" to "the problem of the higher reciprocity laws, handed down from Gauss, Jacobi, Eisenstein and Kummer".