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
• 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
— 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