xvi How to use this text
Figure 1. Diagram of chapter dependencies
characters by means of primitive roots seems less conceptually demanding
than even an epsilon of representation theory. The theorem of Siegel from
Section 7.3 is also needed in Section 9.3. The Borel-Carath´ eodory lemma
of Section 7.4 is also used to prove the Hadamard factorization theorem of
Section 8.6. The Hadamard factorization theorem is needed in Section 9.4
as well as in Chapter 10. As usual, starred material may be omitted without
loss of continuity.
There are exercises at the end of the chapters, for which solutions are not
provided. In addition to the end-of-chapter exercises, there is a chapter with
a selection of other exercises with solutions. As a caution, some exercises
have been marked with a dagger, because they require more work than the
remainder. But they are for the most part not more diﬃcult in the sense of
it being harder to see what to do.
A Summary of Elementary and Algebraic Number Theory with a con-
densed exposition of those concepts on which the book draws is available
on the web. The Summary presupposes familiarity with groups, rings and
fields. All results from elementary and algebraic number theory that are
actually needed in this book are proved in the Summary.