Unlike for local fields, the Galois groups of global fields are not known. Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois group for some number field . Class field theory describes the abelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the Kronecker–Weber theorem, describes the maximal abelian extension of : it is the field obtained by adjoining all primitive th roots of unity. Kronecker's Jugendtraum asks for a similarly explicit description of of general number fields . For imaginary quadratic fields, , , the theory of complex multiplication describes using elliptic curves. For general number fields, no such explicit description is known.
unlike for local fields the galois groups of global fields are not known inverse galois theory studies the (unsolved) problem whether any finite group is the galois group for some number field  class field theory describes the abelian extensions ie ones with abelian galois group or equivalently the abelianized galois groups of global fields a classical statement the kronecker–weber theorem describes the maximal abelian extension of  it is the field obtained by adjoining all primitive th roots of unity kronecker's jugendtraum asks for a similarly explicit description of of general number fields  for imaginary quadratic fields   the theory of complex multiplication describes using elliptic curves for general number fields no such explicit description is known