xvm Prefaces to the German Editions • For understanding the following chapters, the only part of Chap- ter 7 that is necessary is the first part, which deals with the regular heptadecagon (17-gon). • Chapter 8 can be omitted entirely. • In Chapter 9 the set-off sections at the end of the chapter may be skipped. • Chapter 10 and the epilogue may also be omitted. Readers who wish to follow a typical "Algebra I" course should place Chapters 9 and 10, which deal with Galois theory, as well as the epilogue, at the center of their reading. For a deep understanding of the subject the following are of particular importance: the main theorem on symmetric polynomials (Chapter 5), the factorization of polynomials (Chapter 6), and the ideas around cyclotomy (the divi- sion of the circle) (Chapter 7). How much relative attention should be given to the remaining chapters depends on the reader's interests and prior knowledge. Following the historical development of the subject, the presen- tation on the solvability of equations is divided into three parts: • Classical methods of solution, based on more or less complicated equivalent reformulations of equations, were used historically for deriving the general formulas for quadratic, cubic, and quartic equations (Chapters 1 through 3). • Systematic investigation of the discovered solution formulas be- comes possible when one expresses the intermediate results of the individual calculational steps in terms of the totality of the solutions being sought (Chapters 4 and 5). This leads to the solution of equations in special forms, namely, those that are less complex than those in the general form in that they exhibit particular relationships among the solutions that can be formu- lated as polynomial identities. In addition to equations that can be broken down into equations of lower degree (Chapter 6), the so-called cyclotomic equations xn — 1 = 0 are examples of such less-complex equations (Chapter 7). Finally, in this part should be included the attempt, described in Chapter 8, at finding a

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