By Clemens Adelmann
It truly is an old target of algebraic quantity thought to narrate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively defined when it comes to the bottom ?eld. compatible constructions are the top beliefs of the hoop of integers of the thought of quantity ?eld. by means of reading the behaviouroftheprimeidealswhenembeddedintheextension?eld,su?cient info can be accumulated to tell apart the given extension from all different attainable extension ?elds. the hoop of integers O of an algebraic quantity ?eld ok is a Dedekind ring. okay Any non-zero excellent in O possesses hence a decomposition right into a product ok of best beliefs in O that's detailed as much as diversifications of the criteria. This ok decomposition generalizes the best issue decomposition of numbers in Z Z. which will preserve the individuality of the standards, view should be replaced from components of O to beliefs of O . ok okay Given an extension K/k of algebraic quantity ?elds and a main excellent p of O , the decomposition legislation of K/k describes the product decomposition of ok the best generated by way of p in O and names its attribute amounts, i. e. ok the variety of di?erent best perfect elements, their respective inertial levels, and their respective rami?cation indices. Whenlookingatdecompositionlaws,weshouldinitiallyrestrictourselves to Galois extensions. This unique case already o?ers a variety of di?culties.
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