The Arithmetic Problems Of Some Special Algebraic Function Fields

Algebraic number theory, arising from elementary number theory, is concerned with the finite algebraic extensions K of Q, which are called algebraic number fields, and the arithmetic properties of the ring of algebraic integers OK(?)K, defined as the integral closure of Z in K. Similarly, we can consider F(T), the quotient field of F[T], where F is a finite field, and finite algebraic extensions L of F(T). Fields of this type are called algebraic function fields. What's more, an algebraic function fields with a finite constant field is called a global function field. The global function fields are the true analogue of algebraic number fields, and the rich analogy between algebraic number fields and global function fields has been studied for some time. Frequently, the analogy is used to provide insight into the number field realm by formulating suitable analogues of results previously known for function fields. As an example of this statement, the Iwasawa theory for cyclotomic fields was developed in parallel to the theory of constant field extensions for function fields. The analogy is useful in the opposite direction as well. For instance, the theory of the Hilbert Class Fields was thoroughly developed in the number field context before its analogue was studied for function fields. It is well known that we can deal with the problems in global function fields under the aspect of curves over finite fields, in other words, we can choose geometric point to approach this subject. In order to explore the rich analogies that exist between algebraic number fields and global function fields, we approach this theme by very arithmetic method.Now let us describe our main results in this thesis.In chapter1, we first recall some well known facts about algebraic function fields which will be used in following chapters, such as Riemann-Hurwitz formula for algebraic function fields, zeta-function of global function fields and so on. Afterwards, with the help of the expression for zeta function of global function field K, we obtain the formulas for b2g-2(K) and b2g-3(K), where bn(K) denotes the number of effective divisors in K of degree n, and g the genus of K. Using the expression of b2g-2(K) and b2g-3(K), we do some research on the divisibility of class number of constant field extension of global function fields by some special prime number. Finally, the algebraic function fields of the form K(?) are studied in this chapter, and we provide by analyzing the decomposition of primes the explicit genus formula for multiple Kummer function field and composite field of Kummer function field and Artin-Schreier function field.In chapter2, we introduce in section one the genus theory for function fields, and the Conner-Hurrelbrink exact hexagon which is used to study the structure of ideal class group of global function fields in the following section. We present many properties of l-class group of cyclic extensions of prime degree l of global function fields in section two. Using the Conner-Hurrelbrink exact hexagon, we give a new proof of Hasse's result about the l-rank of class group of number fields, and then provide generators of Sylow l-group of ideal class group of function fields. Some relations of class number between biquadratic function fields and their subfields are discussed at the end of this chapter.In chapter3, we first briefly narrate the Drinfeld module theory that developed by V. G. Drinfeld and D. Hayes in the mid-seventies of the last century. Making use of the normalization Drinfeld module theory which was invented by Hayes, we present explicitly the genus formula, which can be considered as the generalization of classical case in cyclotomic function fields, for normalizing field and its cyclotomic extension field. Soon after, we introduce the definition of Carlitz module, a special Drinfeld module, and cyclotomic function fields. Several properties of cyclotomic polynomial in function field context are provided in section two of this chapter. At the end of this chapter, we present the explicit analytic class number formulas for subfields of cyclotomic function field Kp, where K is a rational function field and P is a monic irreducible polynomial in the integral ring of K. Stickelberger-Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial over finite fields. Based on this theorem, we prove in chapter4that every affine polynomial A(x) over F2with degree>1, where A(x)=L(x)+1and L(x)=(?) is a linearized polynomial over F2, is reducible except x2+x+1and x4+x+1. We also give some explicit factors of some special affine pentanomials over F2.