Galois theory occupies a rather central place in modern number theory, from class field theory, to the Langlands program, to the ideas found in Grothendieck's Esquisse d'un Programme. But the Wikipedia article on Galois theory says that the mathematical community of the 19th century was not immediately very receptive to Galois theory. The Kronecker-Weber theorem involves abelian extensions, so would I be correct in assuming that Galois theory had already been adopted by the time of Kronecker? But anyway, my main question is, when did Galois theory start to become so central to number theory?
I suggest that you reed B. Melvin Kiernan's The Development of Galois Theory from Lagrange to Artin. In particular, it says there that:
In the 1890's a few noteworthy expositions of GALOIS Theory were published, perhaps stimulated by the appearance of a German translation of GALOIS' works, in 1889, and the reprinting in book form, in 1897, of GALOlS' papers already published. Certainly the most innovative of these expositions were the two presentations by HEINRICI WEBER (1842–1913), an article, “Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie,” published in 1893, and the development in his Lehrbueh der Algebra, published in 1895.
Note that Kronecker died in 1891. The expositions of Galois theory by Weber were published only after the death of Kronecker. Therefore, it is unlikely that “Galois theory had already been adopted by the time of Kronecker”