scholarly journals Lower bounds on the class number of algebraic function fields defined over any finite field

2012 ◽  
Vol 24 (3) ◽  
pp. 505-540
Author(s):  
Stéphane Ballet ◽  
Robert Rolland
Keyword(s):  
Finite Field ◽  
Lower Bounds ◽  
Class Number ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Function Fields

scholarly journals The distribution of the residues of a quartic polynomial

1967 ◽  
Vol 8 (2) ◽  
pp. 67-88 ◽  
Author(s):  
K. McCann ◽  
K. S. Williams
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Galois Group ◽  
Riemann Hypothesis ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Closure ◽  
Quartic Polynomial ◽  
Algebraic Function Fields ◽  
Image Position

Let f(x) denote a polynomial of degree d defined over a finite field k with q = pnelements. B. J. Birch and H. P. F. Swinnerton-Dyer [1] have estimated the number N(f) of distinct values of y in k for which at least one of the roots ofis in k. They prove, using A. Weil's deep results [12] (that is, results depending on the Riemann hypothesis for algebraic function fields over a finite field) on the number of points on a finite number of curves, thatwhere λ is a certain constant and the constant implied by the O-symbol depends only on d. In fact, if G(f) denotes the Galois group of the equation (1.1) over k(y) and G+(f) its Galois group over k+(y), where k+ is the algebraic closure of k, then it is shown that λ depends only on G(f), G+(f) and d. It is pointed out that “in general”

scholarly journals Algebraic function fields with small class number

1975 ◽  
Vol 7 (1) ◽  
pp. 11-27 ◽  
Author(s):  
James R.C. Leitzel ◽  
Manohar L. Madan ◽  
Clifford S. Queen
Keyword(s):  
Class Number ◽  
Function Fields ◽  
Algebraic Function ◽  
Small Class ◽  
Algebraic Function Fields

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.

scholarly journals Algebraic function fields of class number one

1972 ◽  
Vol 20 (4) ◽  
pp. 423-432 ◽  
Author(s):  
Manohar Madan ◽  
Clifford Queen
Keyword(s):  
Class Number ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Function Fields
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