ABSTRACT
Beyond quadratic extensions, the cyclotomic extensions are the simplest and best understood ones. What makes them even more important is the fact that any Abelian extension is a subextension of a cyclotomic extension. In the next chapter, we shall give a proof of this celebrated theorem. It was stated by Kronecker in 1853 and proved by Weber in 1887. Actually, the first complete proof is due to Hilbert [24]. In 1911, after some unsuccessful attempts, Weber did eventually succeed in providing a correct proof of his own. Several other proofs of this important theorem have appeared in the literature, some only a few years ago.
