Heinrich Martin Weber and David Hilbert created their own class fields in [1891](https://www.sciencedirect.com/science/article/abs/pii/B9780125996617500197) and [1898](https://en.wikipedia.org/wiki/Hilbert_class_field) 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.



Compare with Weber and Hilbert class field
Keith Conrad discusses the history of class fields in these [lecture notes](https://kconrad.math.uconn.edu/blurbs/gradnumthy/cfthistory.pdf). Weber's and Hilbert's definitions are equivalent, but Weber only considered class fields for ideal groups over $\mathbf{Q}$ or imaginary quadratics.


