Acknowledgments xix general solution formula for fifth-degree equations, the result of which is a formula that works only in special cases. • Based on systematic attempts at finding solution formulas, we finally arrive at the limits of solvability of equations in radicals. These limits, as recognized and investigated by Abel and Ga- lois, are dealt with, aside from a brief preview in Chapter 5, in Chapters 9 and 10. The focus here is on Galois groups. With the investigation of Galois groups we reach a level of difficulty well beyond that of the first chapters. Therefore, two different presentations are given. In Chapter 9 a relatively elementary overview is given, supplemented by numerous exam- ples, in which the scope of the concepts introduced is reduced as much as possible. The resulting holes are filled in Chapter 10, which leads to the main theorem of Galois theory, which involves the mathematical objects called fields referred to earlier, which are closed under the four basic arithmetic operations. The dis- cussion of these objects will be limited to those aspects relevant to Galois theory. The reader who wishes to deepen his or her understanding of Ga- lois theory beyond what is contained in this book can move on to any textbook on modern algebra. One might mention as representatives of these books the two classics Algebra, by Bartel Leendert van der Waerden (1903-1996), and Galois theory, by Emil Artin (1898-1962), whose first editions appeared in 1930 and 1948. But conversely, the present book can be seen as an extension of the usual algebra text- books in the direction of providing examples and historical motiva- tion. Acknowledgments I would like to thank all those who shared in the creation of this book: I received considerable advice about errors and infelicities from Jiirgen Behrndt, Rudolf Ketter, and Franz Lemmermeyer. Thanks to their help I was able to reduce the number of errors considerably, though of course the errors that remain are my fault entirely. I thank Vieweg-Verlag and its program director Ulrike Schmickler-Hirzebruch for having accepted this book for publication. Finally, I thank my

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