# Why is the Kronecker-Weber Theorem significant?

I am probably being very naive, but I don't see why the Kronecker-Weber theorem is regarded as such a significant result. Is it surprising and unintuitive? Are many important historical number-theoretic questions answered as immediate corollaries of it?

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To begin with, it is rather surprising and terribly beautiful... –  DonAntonio Feb 23 at 4:59
How so? Perhaps I just don't have enough of an intuition of number fields to appreciate it. –  Mehta Feb 23 at 5:00
How so what , @Mehta ? –  DonAntonio Feb 23 at 5:01

One of the prime directives in all of mathematics is the task of classification, and the classification of various structures or situations that have a given kind of symmetry is no exception. The most canonical choice of field is the field of rational numbers $\Bbb Q$, and so there is the problem of, for instance, finding all number fields with a given Galois group. This is too much to ask in general; the much simpler question "are all finite groups realizeable as symmetries over $\Bbb Q$" is itself a very hard open problem, called the Inverse Galois Problem.
Within the family of isomorphism classes of finite groups, there are various ways to think of it as a heirarchy or a spectrum, where on one end we have those groups that are easiest to work with in a given context, and then on the other extreme there are groups that are hardest. Generally speaking the easiest will be cyclic of prime degree, then all cyclic groups, then abelian, then nilpotent, then solvable, or perhaps only focusing on $p$-groups or on simple groups, etc. So with our Very Hard problem of classifying the Symmetries of Numbers, the first steps on the ladder of difficulty comprise the task of thinking about abelian Galois groups.
Unfortunately, I lack the experience and background to say definitively how KW might be intuitive or why it could be considered surprising. On the former point, perhaps it could be considered analogous to the positive characteristic situation where all abelian extensions of finite fields are contained in cyclotomic extensions $-$ it is much tighter than that though: all extensions of finite fields are cyclic and cyclotomic, so I am skeptical about that thought. On the latter point, as "galois problems" are so apparently hard, it could be considered surprising that such a clean, simple answer turns up for abelian groups, even if abelian is the easy mode part of the game (relatively speaking).