The Distribution of Primes in a Special Ring of Integers

1979 ◽  
Vol 52 (1) ◽  
pp. 31 ◽  
Author(s):  
Rufus Isaacs
Keyword(s):  
Distribution Of Primes ◽  
Ring Of Integers ◽  
Special Ring

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.

scholarly journals Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Riemann Hypothesis ◽  
Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

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.

Sign in / Sign up

Export Citation Format

Share Document