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In this introductory chapter we present three topics. The first one is the basic theory of transcendental fields, which is needed due to the fact that any function field is a finitely generated transcendental extension of a given field.

This chapter will serve as an introduction to our theory of function fields. Using as a source of inspiration compact Riemann surfaces, and especially their fields of meromorphic functions, we first generalize the concept of a function field. In this way we will obtain the general definition of a function field, and establish its most immediate properties.

The Riemann-Roch theorem relates various numbers and invariants of a function field, by means of an equality that plays a central role in our whole theory: It allows us to obtain elements that satisfy given properties, to construct automorphisms or homomorphisms with given characteristics, etc. On the other hand, this equality introduces an arithmetic invariant that is intrinsic to any function field, namely its genus.

In this chapter we present examples that illustrate how one can apply our results of Chapters 2 and 3. We shall first recall a few facts about rational function fields and characterize fields of genus 0.

This chapter is about the Galois theory of function fields.Many of the results presented here are of a general nature, but our interest and emphasis will be focused on function fields.

In this chapter we shall consider congruent function fields, which are fields whose field of constants is finite. This case is the one for which the analogy between number fields and function fields is deepest.

In Chapter 6 we defined the zeta function of a congruence function field. This definition arises from the natural extension of the usual Riemann zeta function $$ G = Aut_k K = \left\{ {\sigma :K \to K\left| {\sigma is{\text{ an automorphism and }}\sigma } \right|_k = Id_k } \right\} $$ . It is known that ζ(s) has a meromorphic extension to the complex plane, with a unique pole at s = 1. This pole is simple with residue 1. Furthermore, ζ(s) has zeros at s = -2 n ( n ζ ℕ) and these are called the trivial zeros of μ( s ). On the other hand, ζ(s) has no zeros different from the trivial ones in ℂ s ≤ ℝe s ≤ 1}. Finally, the Riemann hypothesis states that the zeros of ζ( s ) other than the trivial ones lie on the line of equation ℝe s = 1/2.

We have seen (Remark 5.2.30 and Example 5.2.31) that the field of constants of a constant extension K ℓ can contain ℓ properly. On the other hand, if ℓ is a finite field, the constant field of K ℓ is ℓ (Theorem 6.1.2).

Given a function field K / k , the divisor of any nonzero differential ω has degree 2 gk -2 (Corollary 3.5.5). Consider an extension L /ℓ of K / k ; if we could find a differential Ω of L coming from ω, then we would be able to compare the degrees of Ω and ω, thus obtaining a relation between the respective genera of L and K . In the separable geometric case, we can obtain such a relation between ω and Ω by means of the cotrace of ω, and in this way we get the Riemann-Hurwitz formula.

The term cryptography comes from the two Greek words: kryptós (hidden, secret) and gráphein (to write). In this way, cryptography may be understood as a method of writing in a secret way. More precisely, it is the art of transforming written information from its original or standard form to one that cannot be understood unless one knows a secret key.