XVI Prefaces to the German Editions This abstraction achieved at the beginning of the twentieth cen- tury and today basically unchanged marks both the end of a historical process during which interest in the problem that we have described has shifted in focus: For Cardano and his contemporaries the main problem was to find concrete solutions to explicit problems using pro- cedures of general applicability. But soon the point of view shifted and the focus was on the important properties of the equations. Begin- ning with Galois, but in full force only after the turn of the twentieth century, the focus shifted drastically. Now abstract classes of objects such as groups and fields became the basis for the formulation of a host of problems, including those that inspired the creation of these objects in the first place.2 About This Book. In order to reach as wide an audience as possible (assumed is only general knowledge obtained from college courses in mathematics), no attempt has been made to achieve the level of gen- erality, precision, and completeness that are the hallmarks of mathe- matical textbooks. The focus will be rather on ideas, concepts, and techniques, which will be presented only insofar as they are applicable to some concrete application and make further reading in the exten- sive literature possible. In such a presentation, complicated proofs have no place. However, proofs are without doubt the backbone of any serious engagement with mathematics. In the spirit of compro- mise, difficult proofs, except those in the last chapter, are set off from the main text so that gaps in the logic can be avoided without the flow of the narrative being interrupted. Considerable emphasis is placed on the historical development of the subject, especially since the development of modern mathematics in recent centuries is much less well known than that of the natu- ral sciences, and also because it can be very interesting to be able to give a time-lapse view of false starts and important discoveries. 2 In particular, many important applications have been found in modern infor- mation theory, in particular in cryptography, as in, for example, the public key codes realized in 1978. In these asymmetric encryption procedures, the key for encoding is made public without creating the risk of unauthorized decoding. The mathematical basis for such public key encryption algorithms as RSA and ElGamal is computations carried out in special algebraic objects with a very large—but finite—number of el- ements (precisely, the objects are residue class rings and elliptic curves defined over finite fields). An introduction to this subject can be obtained from Johannes Buch- mann, Introduction to Cryptography, Springer, 2004.

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