Studies On Several Arithmetic Problems Of Global Function Fields

An algebraic function field K with finite constant field is called a global functionfield. There exists t∈K, such that K/Fq(t) is a finite separable extension, where Fq isa finite field with q elements, k = Fq(t) is the rational function field over Fq. Let OKbe the integral closure of the polynomial ring A = Fq[t] in K, and UK be the group ofunits of OK . The following results are obtained.(1) We determine the structure of abelian groups of solutions in OK for the Pellequations x2 ? dy2 = 1(d∈OK \ {0}) .(2) Using Pell equations on A,we give a new proof of the well-known fact that ifthe ideal class number of a real quadratic function field K = k(√D) equals to 1,thenD must be P or QR, where P,Q,R∈A are monic and irreducible ,and Q,R have odddegrees.(3) We obtain two results on the distribution of the values of dth power residuesymbols on A. In particular ,we prove that every nonempty finite subset of A is a setof dth power residue for infinitely many irreducible polynomials in A.(4) Let K be a finite geometric, Galois extension of the rational function field k.For any place v of K , let Kv be the local field at v , Uv be the group of local units at v.A necessary and sufficient condition for the existence of a finite place P of k such thatthe natural map UK/UKd→v|P Uv/Uvd is injective is given here by using C?ebotarev'sdensity theorem.(5) In general Dedekind rings, we introduce the concept of Carmichael ideals ,andprove a generalization of Kolselt's criterion. Suppose OE is any Dedekind ring, E isthe quotient field of OE, we give a necessary and sufficient condition for an ideal inOE generating a Carmichael ideal in L . When E is a function field, this result is muchbetter than the existed result for number field E.(6) For any abelian function field L, we give a lower bound for the Weil heightdepending only on its conductor.(7) We obtain new lower bounds for the sum of degrees of simple and distinctirreducible factors of the polynomial f1 +···+ fn, where fi(1≤i≤n) are pairwise relatively prime polynomials of several variables with coefficients in C.