MONOGENEITY IN CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1589-1607 ◽  
Author(s):  
LEANNE ROBERTSON
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Complex Plane ◽  
Cyclotomic Field ◽  
Difficult Problem ◽  
Cyclotomic Fields ◽  
Ring Extension ◽  
Integral Basis ◽  
Ring Of Integers ◽  
Integral Bases

A number field is said to be monogenic if its ring of integers is a simple ring extension ℤ[α] of ℤ. It is a classical and usually difficult problem to determine whether a given number field is monogenic and, if it is, to find all numbers α that generate a power integral basis {1, α, α2, …, αk} for the ring. The nth cyclotomic field ℚ(ζn) is known to be monogenic for all n, and recently Ranieri proved that if n is coprime to 6, then up to integer translation all the integral generators for ℚ(ζn) lie on the unit circle or the line Re (z) = 1/2 in the complex plane. We prove that this geometric restriction extends to the cases n = 3k and n = 4k, where k is coprime to 6. We use this result to find all power integral bases for ℚ(ζn) for n = 15, 20, 21, 28. This leads us to a conjectural solution to the problem of finding all integral generators for cyclotomic fields.

scholarly journals Power integral bases for real cyclotomic fields

2005 ◽  
Vol 71 (1) ◽  
pp. 167-173 ◽  
Author(s):  
Laurel Miller-Sims ◽  
Leanne Robertson
Keyword(s):  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Integral Basis ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Root Of Unity ◽  
Integral Bases

We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

scholarly journals Modular group representations associated to SO(p)2-TQFTS

2019 ◽  
Vol 28 (05) ◽  
pp. 1950037
Author(s):  
Yilong Wang
Keyword(s):  
Modular Group ◽  
Group Representation ◽  
Cyclotomic Field ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Group Representations ◽  
Weil Representations ◽  
Ring Of Integers ◽  
Integral Bases ◽  
Topological Quantum

In this paper, we prove that for any odd prime [Formula: see text] greater than 3, the modular group representation associated to the [Formula: see text]-topological quantum field theory can be defined over the ring of integers of a cyclotomic field. We will provide explicit integral bases. In the last section, we will relate these representations to the Weil representations over finite fields.

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].

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

Left Cauchy Integral Bases in Linear Topological Spaces

1970 ◽  
Vol 13 (4) ◽  
pp. 431-439 ◽  
Author(s):  
James A. Dyer
Keyword(s):  
Linear Topological Space ◽  
Integral Representations ◽  
Topological Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Natural Analogue ◽  
Cauchy Integral ◽  
Integral Basis ◽  
Integral Bases ◽  
Definition Of

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

scholarly journals Representing Homology Classes on Surfaces

1976 ◽  
Vol 19 (3) ◽  
pp. 373-374 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Unit Circle ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

D(n)-quadruples in the ring of integers of ℚ(√2, √3)

2019 ◽  
Vol 69 (6) ◽  
pp. 1263-1278
Author(s):  
Zrinka Franušić ◽  
Borka Jadrijević
Keyword(s):  
Number Field ◽  
Ring Of Integers

Abstract Let 𝓞𝕂 be the ring of integers of the number field 𝕂 = $\begin{array}{} \displaystyle \mathbb{Q}(\sqrt{2},\sqrt{3}) \end{array}$. A D(n)-quadruple in the ring 𝓞𝕂 is a set of four distinct non-zero elements {z1, z2, z3, z4} ⊂ 𝓞𝕂 with the property that the product of each two distinct elements increased by n is a perfect square in 𝓞𝕂. We show that the set of all n ∈ 𝓞𝕂 such that a D(n)-quadruple in 𝓞𝕂 exists coincides with the set of all integers in 𝕂 that can be represented as a difference of two squares of integers in 𝕂.

scholarly journals Hyponormal Toeplitz operators on H2(T) with polynomial symbols

1996 ◽  
Vol 144 ◽  
pp. 179-182 ◽  
Author(s):  
Dahai Yu
Keyword(s):  
Functional Equation ◽  
Hardy Space ◽  
Closed Form ◽  
Unit Circle ◽  
Toeplitz Operator ◽  
Explicit Expression ◽  
Complex Plane ◽  
Toeplitz Operators ◽  
Computing Process

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

scholarly journals On the isomorphism class of the ring of all integers of a cyclic wildly ramified extension of degree p II

1988 ◽  
Vol 111 ◽  
pp. 165-171 ◽  
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Number Field ◽  
Cyclic Group ◽  
Algebraic Number ◽  
Isomorphism Class ◽  
Algebraic Number Field ◽  
Ring Of Integers

Let k be an algebraic number field with the ring of integers ok = o and let G be a cyclic group of order p, an odd prime.

Trace form associated to cyclic number fields of ramified odd prime degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050080
Author(s):  
Robson R. Araujo ◽  
Ana C. M. M. Chagas ◽  
Antonio A. Andrade ◽  
Trajano P. Nóbrega Neto
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Prime Degree ◽  
Ring Of Integers ◽  
Algebraic Lattices ◽  
Center Density

In this work, we computate the trace form [Formula: see text] associated to a cyclic number field [Formula: see text] of odd prime degree [Formula: see text], where [Formula: see text] ramified in [Formula: see text] and [Formula: see text] belongs to the ring of integers of [Formula: see text]. Furthermore, we use this trace form to calculate the expression of the center density of algebraic lattices constructed via the Minkowski embedding from some ideals in the ring of integers of [Formula: see text].

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