scholarly journals Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1222
Author(s):  
Nicuşor Minculete ◽  
Diana Savin
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Point Of View ◽  
Algebraic Number Fields ◽  
Algebraic Point ◽  
Ring Of Integers ◽  
Euler’S Function

In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

scholarly journals Units in families of totally complex algebraic number fields

2004 ◽  
Vol 2004 (45) ◽  
pp. 2383-2400
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Algebraic Number ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Number Fields ◽  
Imaginary Quadratic Field ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Multidimensional Continued Fraction ◽  
Fundamental Units

Multidimensional continued fraction algorithms associated withGLn(ℤk), whereℤkis the ring of integers of an imaginary quadratic fieldK, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.

scholarly journals Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1710
Author(s):  
Nicuşor Minculete ◽  
Diana Savin
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Generalized Function ◽  
Algebraic Number Fields ◽  
Euler’S Function

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

The isomorphism class of a set of lattices

1989 ◽  
Vol 105 (2) ◽  
pp. 197-210 ◽  
Author(s):  
S. M. J. Wilson
Keyword(s):  
Algebraic Number ◽  
Isomorphism Class ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Galois Modules ◽  
Stable Class ◽  
Residue Fields

AbstractLet R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

scholarly journals A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

scholarly journals K-RATIONAL D-BRANE CRYSTALS

2012 ◽  
Vol 27 (22) ◽  
pp. 1250112
Author(s):  
ROLF SCHIMMRIGK
Keyword(s):  
String Theory ◽  
Algebraic Number ◽  
Special Class ◽  
Building Blocks ◽  
Number Fields ◽  
Central Point ◽  
Algebraic Number Fields ◽  
Special Values ◽  
Toroidal Compactifications ◽  
Base Points

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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