First Steps to Non-Archimedean

There are introduced the most important notions of non-Archimedean (and in particular, p-adic) mathematics and briefly discussed main results which will be used in the next chapters. There is no any possibility for us to present non-Archimedean mathematics (and in particular, non-Archimedean field theory) step by step in rigorous way proving all results which can be useful for us. The only proofs which will be used in further considerations are presented. This non-Archimedean introduction is very brief. The reader who will wish to study this subject more will be able to continue, for example, with the aid of books of Mahler [97] and Schikhov [106]. Mahler and Schikhov proposed different approaches. Mahler was interested very much in non-Archimedean numbers as numbers: digits, expansions, algebraic operations, algorithms of these operations. Schikhov’s book is more useful to study non-Archimedean analysis. Probably, these books is the simplest way for a nonspecialist to begin to work in non-Archimedean.