Residue Free Differentials and the Cartier Operator for
                        Algebraic Function Fields of one Variable

Tetsuo Kodama

doi:10.4153/cjm-1972-090-8

Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-090-8 ◽

1972 ◽

Vol 24 (5) ◽

pp. 905-914

Author(s):

Tetsuo Kodama

Keyword(s):

Function Field ◽

Function Fields ◽

Algebraic Function ◽

Exact Constant ◽

Constant Field ◽

Algebraic Function Field ◽

Algebraic Function Fields ◽

Cartier Operator

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).

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On Rationality of Algebraic Function Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-044-0 ◽

1969 ◽

Vol 12 (3) ◽

pp. 339-341 ◽

Cited By ~ 1

Author(s):

Nobuo Nobusawa

Keyword(s):

Rational Function ◽

Number Field ◽

Algebraic Number ◽

Function Field ◽

Algebraic Function ◽

Algebraic Number Field ◽

Constant Field ◽

Algebraic Function Field ◽

Algebraic Function Fields ◽

Transcendental Extension

Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case.

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DIOPHANTINE DEFINABILITY OVER HOLOMORPHY RINGS OF ALGEBRAIC FUNCTION FIELDS WITH INFINITE NUMBER OF PRIMES ALLOWED AS POLES

International Journal of Mathematics ◽

10.1142/s0129167x98000440 ◽

1998 ◽

Vol 09 (08) ◽

pp. 1041-1066 ◽

Cited By ~ 3

Author(s):

ALEXANDRA SHLAPENTOKH

Keyword(s):

Finite Field ◽

Infinite Number ◽

Function Field ◽

Function Fields ◽

Algebraic Function ◽

Algebraic Function Field ◽

Algebraic Function Fields ◽

Finite Set ◽

Infinite Set

Let K be an algebraic function field over a finite field of constants of characteristic greater than 2. Let W be a set of non-archimedean primes of K, let [Formula: see text]. Then for any finite set S of primes of K there exists an infinite set W of primes of K containing S, with the property that OK,S has a Diophantine definition over OK,W.

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