Application of Kronecker Weber Theorem

Question

I know this may be a very naive question. Please forgive my naivety. What are some of the applications of the Kronecker-Weber Theorem?

Answer 1

To amplify the answer of @Piquito, any time you want an example of an abelian extension of $\Bbb Q$, you can use KW and find one inside a suitable cyclotomic extension.

Want a cyclic extension of degree $3$? There’s one inside each $\Bbb Q(\zeta_n)$ for $n=9$ and any prime $n\equiv1\pmod3$. Want more than one cubic extension inside some field $k\supset\Bbb Q$? Take two or more of the fields mentioned above, and form compositum. For instance $\Bbb Q(\zeta_7,\zeta_9)=\Bbb Q(\zeta_{63})$ has in it four cyclic cubic extensions of $\Bbb Q$.

Of course the real force of KW is that this is the only way to get finite abelian extensions of $\Bbb Q$.

Answer 2

It is one of the most important theorems that exist and, in particular, is useful to characterize Abelian algebraic extensions i.e. those whose Galois group is commutative. In the difficult Class Field Theory, cultivated by few persons, this theorem has a special relief.

Answer 3

This is not an application. But to appreciate the theorem the following points could be handy. Take the two functions $f(x)=x^2$ and $g(x)=x+1$. As highschool algebra tells us that $(x+1)^2\neq x^2 + 1^2$, we see that the two compositions $(f\circ g)(x)$ and $(g\circ f)(x)$ are not the same functions. That is, composition of two functions is rarely commutative.

What is the Galois group of a Galois extension. It consists of field automorphisms of the upper field fixing the lower field. The field automorphisms are first functions (of a very restricted kind). The binary operation of the Galois group is composition of those functions. So the Galois group being abelian means those functions giving field automorphisms commute. That is why so much noise is being made about cyclotomic fileds having abelian Galois groups.

Kronecker-Weber states that these extensions, and the intermediate extensions are the only abelian extensions for the field of the rational numbers. So our luck ran out. We can't find extensions with abelain Galois groups other than the obvious consequent ones from the cyclotomic ones.

Answer 4

A theorem like that of Kronecker and Weber is not measured in terms of applications, it is measured in terms of insight and the potential to generate powerful generalizations. It has given rise to Kronecker's theory of complex multiplications and to one of Hilbert's 23 problems, and is a guiding theorem for classical class field theory. It has various proofs, each of which uses an important technique (Lagrange resolvents and Stickelberger, Hilbert's abelian crossings, decomposition and ramification groups, local fields and a lot more).