An improved method of computing the regulator of a real quadratic function field

1998 ◽  
pp. 607-620 ◽  
Author(s):  
Andreas Stein ◽  
Hugh C. Williams
Keyword(s):  
Function Field ◽  
Quadratic Function ◽  
Improved Method ◽  
Quadratic Function Field ◽  
Real Quadratic

Class Number Divisibility in Real Quadratic Function Fields

1992 ◽  
Vol 35 (3) ◽  
pp. 361-370 ◽  
Author(s):  
Christian Friesen
Keyword(s):  
Class Number ◽  
Function Field ◽  
Class Group ◽  
Quadratic Function ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Function Field ◽  
Quadratic Function Fields ◽  
Real Quadratic ◽  
The Ideal

AbstractLet q be a positive power of an odd prime p, and let Fq(t) be the function field with coefficients in the finite field of q elements. Let denote the ideal class number of the real quadratic function field obtained by adjoining the square root of an even-degree monic . The following theorem is proved: Let n ≧ 1 be an integer not divisible by p. Then there exist infinitely many monic, squarefree polynomials, such that n divides the class number, . The proof constructs an element of order n in the ideal class group.

scholarly journals Pseudo-elliptic integrals, units, and torsion

2005 ◽  
Vol 79 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Francesco Pappalardi ◽  
Alfred J. Van Der Poorten
Keyword(s):  
Function Field ◽  
Function Fields ◽  
Quadratic Function ◽  
Elliptic Integrals ◽  
Square Root ◽  
Base Field ◽  
Quadratic Function Field

AbstractWe remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomialD(x)whose square root generates a quadratic function field with non-trivial unit. We detail the genus I case.

Construction of all cubic function fields of a given square-free discriminant

2015 ◽  
Vol 11 (06) ◽  
pp. 1839-1885 ◽  
Author(s):  
M. J. Jacobson ◽  
Y. Lee ◽  
R. Scheidler ◽  
H. C. Williams
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Number Fields ◽  
Unpublished Manuscript ◽  
Computational Advantage ◽  
Cubic Number Fields ◽  
Quadratic Function Field ◽  
Cubic Function Fields ◽  
Cubic Function ◽  
Real Quadratic

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

2007 ◽  
Vol 59 (1) ◽  
pp. 186-210 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Quadratic Function ◽  
Height Function ◽  
Rational Functions ◽  
Closed Field ◽  
Height Functions ◽  
Planar Curve ◽  
Endomorphism Algebras ◽  
Form Part ◽  
Quadratic Function Field ◽  
Quadratic Function Fields

AbstractPurely simple Kronecker modules ℳ, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h: K ∪ ﹛∞﹜ → ﹛∞, 0, 1, 2, 3, … ﹜ is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X)[Y] called the regulator. When the module ℳ admits nontrivial endomorphisms, f must be linear or quadratic in Y. In that case ℳ is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End ℳ embeds in the quadratic function field K(X)[Y]/(f). For some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module ℳ having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ ﹛∞﹜ and h is finite-valued at least twice on K ∪ ﹛∞﹜. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.

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