scholarly journals Normsets and determination of unique factorization in rings of algebraic integers

1996 ◽  
Vol 124 (6) ◽  
pp. 1727-1732 ◽  
Author(s):  
Jim Coykendall
Keyword(s):  
Unique Factorization ◽  
Algebraic Integers ◽  
Rings Of Algebraic Integers

scholarly journals CHARACTERIZATION OF CODES OF IDEALS OF THE POLYNOMIAL RING F30 2 [x] mod ô€€€ x30 ô€€€ 1 FOR ERROR CONTROL IN COMPUTER APPLICATONS

2016 ◽  
Vol 12 (5) ◽  
pp. 6238-6247
Author(s):  
Maurice Oduor ◽  
Olege Fanuel ◽  
Aywa Shem ◽  
Okaka A Colleta
Keyword(s):  
Polynomial Ring ◽  
Algebraic Number ◽  
Error Control ◽  
Number System ◽  
Code Region ◽  
Unique Factorization ◽  
Algebraic Integers ◽  
Fermat's Last Theorem ◽  
Rings Of Algebraic Integers

The study of ideals in algebraic number system has contributed immensely in preserving the notion of unique factorization in rings of algebraic integers and in proving Fermat's last Theorem. Recent research has revealed that ideals in Noethe-rian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring Fn 2 [x] mod (xn 1) for error control. In this research we generate ideals of the polynomial ring using GAP software and characterize the polycodewords using Shannon's Code region and Manin's bound.

scholarly journals Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

2020 ◽  
Author(s):  
Chris Bruce
Keyword(s):  
Phase Transitions ◽  
Class Group ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
C Algebras ◽  
Rings Of Algebraic Integers ◽  
Beta 2 ◽  
Factor State ◽  
Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

scholarly journals Matrices over rings of algebraic integers

1991 ◽  
Vol 145 ◽  
pp. 1-20 ◽  
Author(s):  
Morris Newman ◽  
Robert C. Thompson
Keyword(s):  
Algebraic Integers ◽  
Rings Of Algebraic Integers

K i Groups of Rings of Algebraic Integers

1975 ◽  
Vol 101 (1) ◽  
pp. 20 ◽  
Author(s):  
B. Harris ◽  
G. Segal
Keyword(s):  
Algebraic Integers ◽  
Rings Of Algebraic Integers
Sign in / Sign up

Export Citation Format

Share Document