Compare with Weber and Hilbert class field

Question

Heinrich Martin Weber and David Hilbert created their own class fields in 1891 and 1897 respectively.

In the past, Weber continued to name $K={Q}(\sqrt{-m}, j(\omega))$, the Kronecker class field of $k={Q}(\sqrt{-m})$, called "species associated with a field k" created by Leopold Kronecker, as 'class field', but later in a series of papers in 1896, Weber defined the concept of his class field. He extended it to the field K related to the congruence class group. Therefore, he used the class field K as a general class field, which is related to the law of decomposition of prime numbers at k.

And Hilbert predicted his own famous class field, the 'Hilbert class field', which was later realized by Philipp Furtwangler.

As a result, My question is what Hilbert thought was lacking in Weber's class field, so there was a need to create a new class field concept.

Answer 1

Hilbert and Weber were more or less working simultaneously and independently on questions that led them to introduce "class fields". Weber was interested in extending Dirichlet's theorem on primes in arithmetic progression to arbitrary number fields; for this he needed the existence of extensions with a precise splitting behavior. He defined "ray class groups" and was able to prove the existence of the corresponding class fields in the cases where the methods of the 19th century were sufficient: for the rationals and complex quadratic number fields (the theory of complex multiplication produces abelian extensions of complex quadratic number fields).

Hilbert, on the other hand, was interested in generalizing reciprocity laws. Kummer had proved the reciprocity law for $p$-th powers only for regular primes, and already Kronecker had pointed out that for closing this gap one had to look at certain unramified abelian extensions. The close connection between the splitting of primes in these extensions and the class group of the base field led Hilbert to his definition of class fields as maximal unramified abelian extensions of number fields. Hilbert proved their existence for number fields with class number $2$; the general case was taken care of by Furtwängler.

Takagi saw that by combining these approaches one would obtain a theory of abelian extensions of arbitrary number fields.

In particular, there was nothing lacking in Weber's notion of class field. It's rather the other way round, as Hilbert only looked at unramified extensions; Hilbert's notion of a class field is a special case of Weber's.

Answer 2

Keith Conrad discusses the history of class fields in these lecture notes. Weber's and Hilbert's definitions are equivalent, but Weber only considered class fields for ideal groups over $\mathbf{Q}$ or imaginary quadratics.