Abstract
Structure and decomposition laws of metabelian extensions of global fields can be studied by applying class field theory for the individual abelian steps. In this way the present paper gives a detailed description of the structure of normal extensions of algebraic number fields, whose group is the dihedral one with order 2 n . At first the algebraic generation of normal fields with dihedral group over an arbitrary field is studied, which gives an explicit decision of the imbedding problem in the simplest cases. In the second section the class field structure of fields with dihedral group is described by idele characters; on this foundation the decomposition laws are formulated.
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