Details

Autor(en) / Beteiligte

• D'MELLO, JOSEPH GERARD

Titel

CLASS GROUPS OF Z(,L)-EXTENSIONS AND SOLVABLE AUTOMORPHISM GROUPS OF ALGEBRAIC FUNCTION FIELDS

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

1983

Link zum Volltext

• ProQuest

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• Let F/K be a field of algebraic functions of one variable having an algebraically closed field K as its field of constants. For a prime l, not necessarily different from the characteristic p of F, let E/K be a cyclic extension of degree l of F/K. Deuring obtained a Riemann-Hurwitz type relation between the l-ranks of the group of divisor classes of degree zero of E and F in the case when E/F is ramified. Safarevic proved this relation in the case l = p. Madan gave a unified proof without imposing any of the previous restrictions. This proof is very much simplified if one makes use of a purely algebraic elementary theorem of Tate. The first chapter of this dissertation uses this idea to generalize the above result to (,l)-extensions, i.e. when K is an infinite l-extension of a finite field. The method of this chapter also yields a proof of a recent result of Kida which expresses a Riemann-Hurwitz type relation between certain of the Iwasawa invariants of fields E and F, where E,F are (,l)-extensions of C-M type (l an odd prime) and E/F is a cyclic extension of degree l. Iwasawa and Sinnott have both given alternate proofs of Kida's theorem. However, the proof in this dissertation is simpler. In the last section of the first chapter, K is an infinite l-extension of a finite field K(,0); F/K(,0) and E/K(,0) are congruence function fields such that E/F is a cyclic extension of prime degree l. Let E(,(INFIN)) = EK and F(,(INFIN)) = FK. Then, assuming that F and F(,(INFIN)) have the same l-rank, we give explicit bounds for the degree l('m) of a constant extension E(,m) of E such that E(,m) and E(,(INFIN)) have the same l-rank, in the ramified as well as in the unramified case. In chapter II, K is an algebraically closed field and x is a transcendental over K. Given a finite group G, does there exist a Galois extension F of K(x) such that Gal(F/K(x)) (TURNEQ) G and Aut(F/K) (TURNEQ) G? L. Greenberg has shown that if K is the field of complex numbers then the above question has an affirmative answer. For an arbitrary algebraically closed field K, Valentini and Madan have shown that the answer is affirmative provided that G is a finite abelian group. The main aim of Chapter II is to extend this result to the class of solvable groups.