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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