1. Concerning These Notes 2 2. Introduction (First Day) 2 3. Algebraic Numbers and Integers 5 4. Linear Algebra for Number Theory 10 5. Review of Field Theory; Traces/Norms 10 6. Units: Some Notes Missing Here 16 7. Diophantine Approximation 16 8. The Trace Pairing 18 9. Linear Algebra and Discriminants 21 10. The Ring of Integers Inside the Number Field 26 11. Some Computational Aspects of Dis...

Abstract Dedekind type DC sums and their generalizations are defined in terms of Euler functions generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind by replacing function appearing sums, they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds new sums. One is unipoly-Dedekind sum associated with 2 unipoly-Euler expres...

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...

In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their “strong analogues”. The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by ...

This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a structure, for monoid domains come here and there. The second part relationship between theory composites, Galois nilpotents. third this shows us some crypto systems. We find generalizations known ciphers taking into account infinite alphabet using simple methods....