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A note on kmodules in an algebraic function field K/k of one variable
Title:  A note on kmodules in an algebraic function field K/k of one variable 
Authors:  Washio, Tadashi 
Authors (alternative):  鷲尾, 忠司 
Issue Date:  29Feb1972 
Publisher:  長崎大学教育学部 
Citation:  長崎大学教育学部自然科学研究報告. vol.23, p.911; 1972 
Abstract:  Let K be an algebraic function field of one variable with a constant field k. It is not necessary, in this note, that k is the exact constant field of K. We shall denote by M a finitely generated kmodule in K. Moreover n(M) denotes the denominator divisor of M, i.e., n(M) is the divisor of K defined by ord p n(M)=max{ord p n(x)} x∈M xキO for every prime divisor p of K, where ord denotes the order at p and n(x) means the denominator divisor of x. Then it is wellknown that there exists some element x in M such that n(M)=n(x), if k contains enough elements. (e.g., E. Artin 〔1〕; P.318, Lemma 2). The purpose of this note is to study it in minute detail. Under the condition d(n(M))≦ k, we shall prove that there exists an element x in M such that n(M)=n(x) in §2, where d(n(M)) means the degree of n(M) and k denotes the number of all the elements contained in k; if k is not finite, then we shall put k=∞. In §3, we shall show that the above inequality is the best condition in a sense. 
URI:  http://hdl.handle.net/10069/33016 
ISSN:  0386443X 
Type:  Departmental Bulletin Paper 
Text Version:  publisher 
Appears in Collections:  No. 23

Citable URI :
http://hdl.handle.net/10069/33016

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