It is an historical goal of algebraic number theory to relate all algebraic extensions of a number field in a unique way to structures that are exclusively described in terms of the base field. Suitable structures are the prime ideals of the ring of integers of the considered number field. By examining the behaviourof the prime ideals when embedded in the extension field, sufficient information should be collected to distinguish the given extension from all other possible extension fields. The ring of integers O of an algebraic number field k is a Dedekind ring. k Any non-zero ideal in O possesses therefore a decomposition into a product k of prime ideals in O which is unique up to permutations of the factors. This k decomposition generalizes the prime factor decomposition of numbers in Z Z. In order to keep the uniqueness of the factors, view has to be changed from elements of O to ideals of O . k k Given an extension K/k of algebraic number fields and a prime ideal p of O , the decomposition law of K/k describes the product decomposition of k the ideal generated by p in O and names its characteristic quantities, i. e. K the number of different prime ideal factors, their respective inertial degrees, and their respective ramification indices. When looking at decomposition laws, we should initially restrict ourselves to Galois extensions. This special case already offers quite a few difficulties.