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

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.

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

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.

The Inverse Problem Associated to the Davenport Constant for $C_2\oplus C_2 \oplus C_{2n}$, and Applications to the Arithmetical Characterization of Class Groups

The inverse problem associated to the Davenport constant for some finite abelian group is the problem of determining the structure of all minimal zero-sum sequences of maximal length over this group, and more generally of long minimal zero-sum sequences. Results on the maximal multiplicity of an element in a long minimal zero-sum sequence for groups with large exponent are obtained. For groups of the form $C_2^{r-1}\oplus C_{2n}$ the results are optimal up to an absolute constant. And, the inverse problem, for sequences of maximal length, is solved completely for groups of the form $C_2^2 \oplus C_{2n}$. Some applications of this latter result are presented. In particular, a characterization, via the system of sets of lengths, of the class group of rings of algebraic integers is obtained for certain types of groups, including $C_2^2 \oplus C_{2n}$ and $C_3 \oplus C_{3n}$; and the Davenport constants of groups of the form $C_4^2 \oplus C_{4n}$ and $C_6^2 \oplus C_{6n}$ are determined.

GAUSS AND KLOOSTERMAN SUMS OVER RESIDUE RINGS OF ALGEBRAIC INTEGERS

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with [Formula: see text] as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp (2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum [Formula: see text] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums [Formula: see text] are well-defined for any numerical character η ( mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2.

Euclidean Rings of Algebraic Integers

Let K be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

Existential definability with bounds on archimedean valuations

We show that a solution to Hilbert's Tenth Problem in the rings of algebraic integers and bigger subrings of number fields where it is currently not known, is equivalent to a problem of bounding archimedean valuations over non-real number fields.