Computation of K2 for the ring of integers of quadratic imaginary fields

2001 ◽  
Vol 44 (7) ◽  
pp. 846-855
Author(s):  
Sheng Chen ◽  
Hong You
Keyword(s):  
Ring Of Integers ◽  
Quadratic Imaginary Fields

Several Diophantine Equations in Some Rings of Integers of Quadratic Imaginary Fields

2007 ◽  
Vol 14 (04) ◽  
pp. 661-668 ◽  
Author(s):  
Kejian Xu ◽  
Yongliang Wang
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Ring Of Integers ◽  
Rings Of Integers ◽  
Quadratic Imaginary Fields ◽  
Fermat Equation

In this paper, it is proved that the Diophantine equation x4-y4 =z2 has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields [Formula: see text] for d=11, 19, 43, 67, 163, which implies that the Fermat equation x4+y4 =z4 has no non-trivial solutions in these fields either. Then all the solutions of the Pocklington equation x4-x2y2+y4 =(-1)σz2 (σ =0 or 1) in the ring of integers of [Formula: see text] are determined, and as an application, the result is applied to K2 of a field.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

96.36 Prime and irreducible elements of the ring of integers modulo n

2012 ◽  
Vol 96 (536) ◽  
pp. 283-287
Author(s):  
M. H. Jafari ◽  
A. R. Madadi
Keyword(s):  
Ring Of Integers ◽  
Irreducible Elements

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 On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field

2021 ◽  
Author(s):  
A. Haddley ◽  
R. Nair
Keyword(s):  
Continued Fraction ◽  
Residue Class ◽  
Natural Extension ◽  
Abelian Topological Group ◽  
Ring Of Integers ◽  
Residue Class Field ◽  
Finite Set ◽  
Compact Abelian Topological Group ◽  
Invertible Elements ◽  
Average Behaviour

AbstractLet $${\mathcal {M}}$$ M denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $$k^{\times }$$ k × , and a uniformizer we denote $$\pi $$ π . In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$ T v : M → M defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ T v ( x ) = π v ( x ) x - b ( x ) , where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$ π v ( x ) x belongs in $$k^{\times }$$ k × . We show that $$T_v$$ T v preserves Haar measure $$\mu $$ μ on the compact abelian topological group $${\mathcal {M}}$$ M . Let $${\mathcal {B}}$$ B denote the Haar $$\sigma $$ σ -algebra on $${\mathcal {M}}$$ M . We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$ ( M , B , μ , T v ) is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$ # ( k ) # ( k × ) log ( # ( k ) ) . The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$ T v . Here for a finite set A its cardinality has been denoted by $$\# (A)$$ # ( A ) . In the case $$K = {\mathbb {Q}}_p$$ K = Q p , i.e. the field of p-adic numbers, the map $$T_v$$ T v reduces to the well-studied continued fraction map due to Schneider.

The semigroup of endomorphisms of a Boolean ring

1970 ◽  
Vol 11 (4) ◽  
pp. 411-416 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Rational Numbers ◽  
Boolean Ring ◽  
Ring Of Integers ◽  
The Family ◽  
Identity Automorphism

The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

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