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.

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.

A note on the integral trace form in cyclotomic fields

2015 ◽  
Vol 14 (04) ◽  
pp. 1550045 ◽  
Author(s):  
J. Carmelo Interlando ◽  
Trajano Pires da Nóbrega Neto ◽  
Tatiana Miguel Rodrigues ◽  
José Othon Dantas Lopes
Keyword(s):  
Nonzero Element ◽  
Cyclotomic Field ◽  
Explicit Description ◽  
Complex Conjugate ◽  
Cyclotomic Fields ◽  
Trace Form ◽  
Root Of Unity

Let m ≥ 3 be an integer, ζm ∈ ℂ a primitive mth root of unity, and Km the cyclotomic field ℚ(ζm). An explicit description of the integral trace form [Formula: see text] where [Formula: see text] is the complex conjugate of x is presented. In the case where m is prime, a procedure for finding the minimum of the form subject to x being a nonzero element of a certain ℤ-module in ℤ[ζm] is presented.

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.

scholarly journals Trigonometric identities

1986 ◽  
Vol 9 (4) ◽  
pp. 705-714
Author(s):  
Malvina Baica
Keyword(s):  
Positive Integer ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Root Of Unity ◽  
Trigonometric Identities

In this paper the author obtains new trigonometric identities of the form2(p−1)(p−2)2∏k=1p−2(1−cos2πkp)p−1−k=pp−2which are derived as a result of relations in a cyclotomic fieldℛ(ρ), whereℛis the field of rationals andρis a root of unity.Those identities hold for every positive integerp≥3and any proof avoiding cyclotomic fields could be very difficult, if not insoluble. Two formulas∑k=1p−12(−1)(p2k)tanp−1−2kϕ=0  and−1+∑k=0p−12(−1)k(∑i=0p−1−2k2(p2k+2i)(k+1k))cosp−2kϕ=0stated only by Gauss in a slightly different form without a proof, are obtained and used in this paper in order to give some numeric applications of our new trigonometric identities.

Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

1983 ◽  
Vol 26 (4) ◽  
pp. 464-472 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Lower Bounds ◽  
Class Number ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Asymptotic Results ◽  
Root Of Unity ◽  
Almost All

AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n. Also for those Kn and with a given number of ramified primes, we obtain lower bounds for certain types of densities for m dividing hn and .

scholarly journals Integral bases of pure fields with square-free parameter

2020 ◽  
Vol 57 (1) ◽  
pp. 91-115
Author(s):  
László Remete
Keyword(s):  
Free Parameter ◽  
Period Length ◽  
Quadratic Fields ◽  
Integral Basis ◽  
Algebraic Integers ◽  
Free Part ◽  
Integral Bases ◽  
Special Cases ◽  
Ring Of Algebraic Integers ◽  
Quadratic Case

Abstract Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases. In this paper we explicitly give an integral basis of the field , where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of is repeating periodically in m with period length depending on n.

AN EFFICIENT SEVENTH POWER RESIDUE SYMBOL ALGORITHM

2010 ◽  
Vol 06 (08) ◽  
pp. 1831-1853 ◽  
Author(s):  
PERLAS C. CARANAY ◽  
RENATE SCHEIDLER
Keyword(s):  
Cyclotomic Field ◽  
Jacobi Symbol ◽  
Computing Power ◽  
Efficient Procedure ◽  
Ring Of Integers ◽  
Public Key Cryptosystems ◽  
Root Of Unity ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Crucial Ingredient

Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7th power residue symbols. The method employs arithmetic in the field ℚ(ζ), with ζ a primitive 7th root of unity, and its ring of integers ℤ[ζ]. We give an explicit characterization for an element in ℤ[ζ] to be primary, and provide an algorithm for finding primary associates of integers in ℤ[ζ]. Moreover, we formulate explicit forms of the complementary laws to Kummer's 7th degree reciprocity law, and use Lenstra's norm-Euclidean algorithm in the cyclotomic field.

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.

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.

The Class Number of the Cyclotomic Field

1951 ◽  
Vol 3 ◽  
pp. 486-494 ◽  
Author(s):  
N. C. Ankeny ◽  
S. Chowla
Keyword(s):  
Class Number ◽  
Cyclotomic Field ◽  
Root Of Unity

Let g denote an odd prime, and h = h(g) the class number of the cyclotomic field R(), where is a primitive gth root of unity. It is known that we can write

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