By Clemens Adelmann
It truly is an historic target of algebraic quantity concept to narrate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively defined when it comes to the bottom ?eld. compatible buildings are the best beliefs of the hoop of integers of the thought of quantity ?eld. by way of reading the behaviouroftheprimeidealswhenembeddedintheextension?eld,su?cient info could be accrued to differentiate the given extension from all different attainable extension ?elds. the hoop of integers O of an algebraic quantity ?eld ok is a Dedekind ring. ok Any non-zero excellent in O possesses for this reason a decomposition right into a product ok of major beliefs in O that is distinctive as much as variations of the criteria. This ok decomposition generalizes the best issue decomposition of numbers in Z Z. so that it will preserve the distinctiveness of the criteria, view needs to be replaced from parts of O to beliefs of O . okay ok Given an extension K/k of algebraic quantity ?elds and a primary perfect p of O , the decomposition legislations of K/k describes the product decomposition of okay the correct generated by way of p in O and names its attribute amounts, i. e. ok the variety of di?erent leading excellent components, their respective inertial levels, and their respective rami?cation indices. Whenlookingatdecompositionlaws,weshouldinitiallyrestrictourselves to Galois extensions. This precise case already o?ers a variety of di?culties.
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