Sunghan Bae

Cyclotomic units and Stickelberger ideals of global function fields

In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.

Class numbers of some abelian extensions of rational function fields

Let P be a monic irreducible polynomial. In this paper we generalize the determinant formula for h(K+pn) of Bae and Kang and the formula for h-(Kpn) of Jung and Ahn to any subfields K of the cyclotomic function field Kpn. By using these formulas, we calculate the class numbers h-(K), h(k+) of all subfields K of Kp when q and deg(P) are small.

Determinant formulas for class numbers in function fields

In this paper, by extending Kuceras idea to the function field case, we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields. We also provide several examples using these determinant formulas.

ℓ-RANKS OF CLASS GROUPS OF FUNCTION FIELDS

In this paper we give asymptotic formulas for the number of l-cyclic extensions of the rational function eld k = F q(T ) with pre- scribed l-class numbers inside some cyclotomic function elds, and den- sity results for l-cyclic extensions of k with certain properties on the ideal class groups.

Class number parities of compositum of quadratic function fields

Abstract The parities of ideal class numbers of compositum of quadratic function fields are studied. Especially, the parities of ideal class numbers of Fq(t)(P,Q) and Fq(t)(P,Q,R)

On normalized generating sets for GQC codes over Z2

\mathbb{F}_q(t)(\sqrt P,\sqrt Q) \text{ and } \mathbb{F}_q(t)(\sqrt P,\sqrt Q,\sqrt R)

CLASS NUMBER DIVISIBILITY OF QUADRATIC FUNCTION FIELDS IN EVEN CHARACTERISTIC

are completely determined in detail, where P,Q,R are monic irreducible polynomials of even degrees.

4-Ranks of class groups of quadratic extensions of certain quadratic function fields

Abstract Let r i be positive integers and R i = Z 2 [ x ] / 〈 x r i − 1 〉 for 1 ≤ i ≤ l . Denote R = R 1 × R 2 × ⋯ × R l . Generalized quasi-cyclic (GQC) code C of length ( r 1 , r 2 , … , r l ) over Z 2 can be viewed as Z 2 [ x ] -submodule of R . In this paper, we investigate the algebraic structure of C by presenting its normalized generating set. We also present a method to determine the normalized generating set of the dual code of C , which is derived from the normalized generating set of C .

Class numbers of cyclotomic function fields

Abstract. We find a lower bound on the number of real/inert imagi-nary/ramified imaginary quadratic extensions of the function field F q (t)whose ideal class groups have an element of a fixed order, where q is apower of 2. 1. IntroductionLet k = F q (t) be the rational function field over the finite field F q andA= F q [t]. Let ∞ be the infinite place of k associated to (1/t). Throughout thepaper, by a quadratic function field, we always mean a quadratic extension ofk. A quadratic function field F is said to be real if ∞ splits in F, and imaginaryotherwise. Assume that q is odd. Then any quadratic function field F can bewritten as F = k(√D), where D is a square-free polynomial in A. Let O F bethe integral closure of Ain F. In [2], Murty and Cardon proved that there are≫ q l( 12 + 1g ) imaginary quadratic function fields F = k(√D) such that degD ≤ land the ideal class group of O F has an element of order g. This result is thefunction field analogue of the result of Murty for imaginary quadratic fields([5]). In [4], Friesen proved the existence of infinitely many real quadraticfunction fields F whose ideal class numbers are divisible by a given positiveinteger g. In [3], using the Friesen’s result, Chakraborty and Mukhopadhyayproved that there are ≫ q

Anderson's double complex and gamma monomials for rational function fields

We obtain some density results for the 4-ranks of class groups of quadratic extensions of quadratic function fields analogous to those results of F. Gerth in the classical case.