On Rationality of Algebraic Function Fields

1969 ◽  
Vol 12 (3) ◽  
pp. 339-341 ◽  
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.

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

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).

scholarly journals On the Galois Cohomology Group of the Ring of Integers in a Global Field and its Adele Ring

1968 ◽  
Vol 32 ◽  
pp. 247-252 ◽  
Author(s):  
Yoshiomi Furuta ◽  
Yasuaki Sawada
Keyword(s):  
Algebraic Number ◽  
Cohomology Group ◽  
Present Note ◽  
Algebraic Function ◽  
Galois Cohomology ◽  
Algebraic Number Field ◽  
Global Field ◽  
Constant Field ◽  
Algebraic Function Field ◽  
Ring Of Integers

By a global field we mean a field which is either an algebraic number field, or an algebraic function field in one variable over a finite constant field. The purpose of the present note is to show that the Galois cohomology group of the ring of integers of a global field is isomorphic to that of the ring of integers of its adele ring and is reduced to asking for that of the ring of local integers.

Decidable fragments of field theories

1990 ◽  
Vol 55 (3) ◽  
pp. 1007-1018 ◽  
Author(s):  
Shih-Ping Tung
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Field Theories ◽  
Extension Field ◽  
Radical Extension ◽  
Transcendental Extension

AbstractWe say φ is an ∀∃ sentence if and only if φ is logically equivalent to a sentence of the form ∀x∃yψ(x, y), where ψ(x, y) is a quantifier-free formula containing no variables except x and y. In this paper we show that there are algorithms to decide whether or not a given ∀∃ sentence is true in (1) an algebraic number field K, (2) a purely transcendental extension of an algebraic number field K, (3) every field with characteristic 0, (4) every algebraic number field, (5) every cyclic (abelian, radical) extension field over Q, and (6) every field.

scholarly journals Nonstandard Arithmetic of Polynomial Rings

1987 ◽  
Vol 105 ◽  
pp. 33-37 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Rational Function ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Polynomial Rings ◽  
Finite Degree

Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima provedTHEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such thatF(g(T1,…,Tm), T1,…,T)= 0.

scholarly journals Linear Dynamical Systems of Higher Genus

1994 ◽  
Vol 1 (5) ◽  
pp. 505-521
Author(s):  
V. Lomadze
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Rational Function ◽  
Function Field ◽  
Algebraic Function ◽  
Linear Dynamical Systems ◽  
Algebraic Function Field ◽  
Rational Function Field ◽  
Higher Genus

Abstract A class of linear systems which after ordinary linear systems are in a certain sense the simplest ones, is associated with every algebraic function field. From the standpoint developed in the paper ordinary linear systems are associated with the rational function field.

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

1998 ◽  
Vol 09 (08) ◽  
pp. 1041-1066 ◽  
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.

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Juraj Kostra
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Basis ◽  
Prime Degree ◽  
One To One

AbstractLet K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

Sign in / Sign up

Export Citation Format

Share Document