  The aim of this book is to introduce the reader to the beauty of Algebra,
through a journey from the natural numbers to prime fields and finite fields,
with some detours.
  Many books are devoted to the construction of these fields from the natural
numbers. Perhaps the first one is [Landau 1930], see also the books [Ebbinghaus
et.al. 1991], [Lay 2005], and the references therein. An important step in this
process is the construction of the field of rational numbers. It turns out that
this field is the only (in an appropriate sense) infinite prime field.
Motivated by the reading of [Loonstra 1972], I decided to write an essentially
self-contained book devoted to the construction not only of the field of
rational numbers but of all prime fields and, more interestingly, to the proof
of existence and "uniqueness" of all finite fields. Only a knowledge of basic
set theory and some familiarity with mathematical reasoning (see for example
[11, Chapters 1 and 2]) are assumed.
