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Power Bases for Cyclotomic Integer Rings
Abstract
Letpbe an odd prime and pbe the ring of integers in the cyclotomic fieldQ(ζ), whereζis a primitivepth root of unity. Then p=Z[α] ifα=ζ, 1/(1+ζ), or one of the conjugates of these two elements. In 1988, Bremner [3] conjectured that up to integer translation there are no further generators for pand proved that this is indeed the case whenp=7. We establish a criterion for verifying Bremner's conjecture for a given regular primepand use it to prove the conjecture forp⩽23,p≠17. A key step in the proof of the criterion is a determinant formula for the relative class numberh−ofQ(ζ).
... Cyclotomic fields are an interesting case because power integral bases always exist and in some cases we can find all the generators. See Bremner [1] and Robertson [9] for a study of power integral bases in prime cyclotomic fields. See Robertson [10] for the determination of all power integral bases in 2-power cyclotomic fields. ...
... This conjecture is due to Bremner [1] in the case m = 1, and he proves the conjecture for q = 7. Bremner's conjecture has been verified for p 23 [9,14]. Here we study power integral bases and Conjecture 1.3 when m > 1. ...
... In this section we briefly discuss computational results on class numbers of real cyclotomic fields Q(ζ + ¯ ζ ), and the likelihood that the class number hypothesis in Theorem 1.1 is satisfied, that is that gcd(h + q , p(p − 1)/2) = 1 for q = p m . In the case where ζ is a pth root of unity, it was shown in Robertson [9] that if Z[α] = Z[ζ ] and α is not equivalent to ζ , then α + ¯ α is an odd integer. Thus, the class number hypothesis is not necessary for m = 1. ...
Let p be an odd prime and q=pm, where m is a positive integer. Let ζ be a primitive qth root of unity, and Oq be the ring of integers in the cyclotomic field Q(ζ). We prove that if Oq=Z[α] and gcd(hq+,p(p−1)/2)=1, where hq+ is the class number of Q(ζ+ζ−1), then an integer translate of α lies on the unit circle or the line Re(z)=1/2 in the complex plane. Both are possible since Oq=Z[α] if α=ζ or α=1/(1+ζ). We conjecture that, up to integer translation, these two elements and their Galois conjugates are the only generators for Oq, and prove that this is indeed the case when q=25.
... As is well known, Hasse's problem to characterize whether the ring Z K of integers in a field K is of monogenesis or not is treated by Dummit and Kisilevsky [1], Gras [4], Huard et al. [7], Robertson [10], Schertz [13], The´rond [15] and others. Gaa´l et al. and Gy + o ory gave algorithm for determining the power bases of the rings in certain algebraic number fields and several monogenic examples [2,3,6]. ...
... In two cases of the maximal imaginary sextic subfields K of conductors 28 and 36 in k 28 and k 36 ; the proof of Theorem 1(2) involves that there are generators AEiS; AEi=S and their conjugates only for Z K except for the parallel transformations of them by rational integers. For the cases of cyclotomic fields k p of the prime conductor p; p423; pa17; Robertson completely determined the generators of Z k p in [10]. ...
Let K be the composite field of an imaginary quadratic field Q(ω) of conductor d and a real abelian field L of conductor f distinct from the rationals Q, where (d,f) = 1. Let ZK be the ring of integers in K. Then concerning to Hasse's problem we construct new families of infinitely many fields K with the non-monogenic phenomena (1), (2) which supplement (J. Number Theory 23 (1986), 347-353; Publ. Math. Fac. Sci Besançon, Theor. Nombres (1984) 25pp) and with monogenic (3). (1) If Q(ω) ≠ the Gauβ field Q(i), then ZK is ofnon-monogenesis. (2) If Q(ω) = Q(i), then for a sextic field K, ZK is of non-monogenesis except for two fields K of conductors 28 and 36. (3) Let Q(ω) = Q(i). If ZK has a power basis, then ZL must have a power basis. Conversely, let L be the maximal real subfield kf+ of a cyclotomic field kf, namely K be the maximal imaginary subfield of k4f of conductor 4f. Then ZK has a power basis.
... In Robertson [15] we consider the problem of determining all power bases for prime cyclotomic fields. Let`pLet`Let`p be a primitive pth root of unity. ...
... Let`pLet`Let`p be a primitive pth root of unity. One of the results in [15] is the following: If Z[:]=Z[` p ] and : is not Z-equivalent tò p then :+:Ä is equal to an odd integer. That is, up to Z-equivalence, : either lies on the unit circle or on the line Re(z)=1Â2, z # C. Our next theorem shows that the situation is very different for 2-power cyclotomic fields. ...
Let ζ be a primitive 2mth root of unity. We prove that Z[α]=Z[ζ] if and only if α=n±ζi for some n, i∈Z, i odd. This is the first example of number fields of arbitrarily large degree for which all power bases for the ring of integers are known.
... From the theorem we see that if an element θ ∈ Q(ζ k ) can be found such that ζ k = u(θ), u(x) ∈ Q[x], then Φ k (u(x)) is factorable. Definition 3. [23] Let K be a number field and O K is its ring of integers, then O K is said to have a power integral basis if there exists an element α of O K such that ...
... By Theorem 1, Φ k (u(x)) splits. We used the results of [23, 24, 13] to find elements that generate a power integral basis for Q(ζ k ). If k = p or p m , where p is a prime, we choose ...
In constructing pairing-friendly ellitpic curves, the curve parameters are often represented by polynomials with rational coefficients. For efficiently generating the curves parameters, the degree of the complex multipliction polynomial must be less than 3. A method is proposed to constructed suitable polynomials which will make the degree of the complex multipliction polynomial less than 3. Some examples are given, especially when embedding degree is 8. It is generally believed that when embedding degree is 8 the complex multipliction polynomial whose degree is less than 3 does not exist.
... In the cubic case, however, these studies begin to get more complicated. In fact there are an infinite number of cyclic cubic fields which have a power basis and also an infinite number which do not, and similarly for quartic fields [44]. ...
Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase, we propose the definition of a parity-check matrix and Tanner graph for Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-BF channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order n. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. We provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in BF channels. We generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields.
... In the cubic case, however, these studies begin to get more complicated. In fact there are an infinite number of cyclic cubic fields which have a power basis and also an infinite number which do not, and similarly for quartic fields [35]. ...
LDPC lattices were the first family of lattices which have an efficient decoding algorithm in high dimensions over an AWGN channel. Considering Construction D' of lattices with one binary LDPC code as underlying code gives the well known Construction A LDPC lattices or 1-level LDPC lattices. Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as block-fading channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, have been generalized to a lattice construction from totally real and complex multiplication fields. This generalized Construction A of lattices provides signal space diversity intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper we construct full diversity LDPC lattices for block-fading channels using Construction A over totally real number fields. We propose a new iterative decoding method for these family of lattices which has complexity that grows linearly in the dimension of the lattice. In order to implement our decoding algorithm, we propose the definition of a parity check matrix and Tanner graph for full diversity Construction A lattices. We also prove that the constructed LDPC lattices together with the proposed decoding method admit diversity order n-1 over an n-block-fading channel.
... Proof. By [28] up to equivalence all generators of power integral bases of Z[ζ] are ζ and 1 1+ζ . One immediately checks that f k (X) = µ ζ (X + k) ∈ K ⇐⇒ k ≥ 4, hence k ζ = 4. ...
,Canonical number,systems,can be viewed,as natu- ral generalizations of radix representations,of ordinary integers to algebraic integers. A slightly modified,version of an algorithm,of B. Kovács and A. Pethfi is presented here for the determination of canonical number,systems in orders of algebraic number,fields.
... Cyclotomic fields are an interesting case because power integral bases always exist and in some cases we can find all the generators (see Nagell [9], Bremner [1], and Robertson [10,11]). Real cyclotomic fields (that is, the maximal real subfields of cyclotomic fields) are also interesting because again power integral bases always exist. ...
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 ℤ[α]=ℤ[ζ+ζ -1 ] if α=ζ+ζ -1 , 1/(ζ+ζ -1 ), 1/(ζ+ζ -1 +1), 1/(ζ+ζ -1 -1), 1/(ζ+ζ -1 +2) 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 generalize to an additional generator for 13<p<1000.
... De plus on remarque que ζ p n'est pas équivalent à ω si p > 3 et ω + ω = 1. Bremner [1] conjecture qu'il n'existe pas d'autres classes de générateurs de Z[ζ p ] , plus précisément : Robertson [7] donne une réponse partielle à la conjecture de Bremner : La conjecture 1 se généralise aux corps cyclotomiques engendrés par une racine q-ième de l'unité, où q est la puissance d'un nombre premier, de la façon suivante : soit α ∈ Z[ζ q ] tel que Z[α] = Z[ζ q ]. Alors soit α est équivalent à ζ q , soit α est équivalent à 1/(ζ q + 1). ...
Let p be an odd prime and q=pm, where m is a positive integer. Let ζq be a qth primitive root of 1 and Oq be the ring of integers in Q(ζq). In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372–384] I. Gaál and L. Robertson show that if (hq+,p(p−1)/2)=1, where hq+ is the class number of Q(ζq+ζq¯), then if α∈Oq is a generator of Oq (in other words Z[α]=Oq) either α is equals to a conjugate of an integer translate of ζq or α+α¯ is an odd integer. In this paper we show that we can remove the hypothesis over hq+. In other words we show that if α∈Oq is a generator of Oq then either α is a conjugate of an integer translate of ζq or α+α¯ is an odd integer.
Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as BF channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these family of lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for full-diversity algebraic LDPC lattices which are generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase of our decoding algorithm, we propose the definition of a parity-check matrix and Tanner graph for full-diversity algebraic Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-block-fading channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order
$n$
over an
$n$
-block-fading channel. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. In contrast with the known results on AWGN channels in which non-binary Construction A lattices always outperform the binary ones, we provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in block fading channels. Since available lattice construction methods from totally real and complex multiplication (CM) fields do not provide diversity in the binary case, we generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields.
In this chapter we give applications of former results to some types of higher degree fields. Some of these applications concern infinite parametric families of fields.
In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of their multiplicative independence. In the last part we state an appropriate variant of Cobham’s theorem.
Bettale, Faugère, and Perret [3] present and analyze a hybrid method for solving multivariate polynomial systems over finite fields that mixes Gröbner bases computations with an exhaustive search. Inspired by their method, we use a hybrid approach to characterize all power integral bases in the pth cyclotomic field Q(ζp) for the regular primes p = 29, 31, 41. For each prime p this involves solving a system of (p−1)/2 multivariate polynomial equations of degree (p−1)/2 in (p−1)/2 variables over the finite field Z/pZ.
Let k be a finite extension of ℚ and L be an extension of k with rings of integers O k and O L , respectively. If O L =O k [θ], for some θ in O L , then O L is said to have a power basis over O k . In this paper, we show that for a Galois extension L/k of degree p m with p prime, if each prime ideal of k above p is ramified in L and does not split in L/k and the intersection of the first ramification groups of all the prime ideals of L above p is non-trivial, and if p-1∤2[k:ℚ], then O L does not have a power basis over O k . Here, k is either an extension with p unramified or a Galois extension of ℚ, so k is quite arbitrary. From this, for such a k the ring of integers of the nth layer of the cyclotomic ℤ p -extension of k does not have a power basis over O k , if (p,[k:ℚ])=1. Our results generalize those by J. J. Payan [Ark. Math. 11, 239–244 (1973; Zbl 0269.12003)] and M. Horinouchi [‘On power bases of the ring of integers for a cyclic extension of prime degree l over a real quadratic field which is unramified outside prime ideals of k above l’ (Japanese), Master’s Thesis (2011)], who treated the case k a quadratic number field and L a cyclic extension of k of prime degree. When k=ℚ, we have a little stronger result.
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.
Let L be a number field and let 𝒪L be its ring of integers. It is a very difficult problem to decide whether 𝒪L has a power basis. Moreover, if a power basis exists, it is hard to find all the generators of 𝒪L over ℤ. In this paper, we show that if α is a generator of the ring of integers of an abelian imaginary field whose conductor is relatively prime to 6, then either
α is an integer translate of a root of unity, or is an odd integer. From this result and other remarks it follows that if β is a generator of the ring of integers of an abelian imaginary field with conductor relatively prime to 6 and β is not an integer translate of a root of unity, then is a generator of the ring of integers of the maximal real field contained in ℚ(β). Finally, we use a result of Gras to prove that if d > 1 is an integer relatively prime to 6, then all but finitely many imaginary extensions of ℚ of degree 2d have a ring of integers that does not have a power basis.
We explain why it is reasonable to conjecture that if is a totally imaginary quartic unit, then is in general a fundamental unit of the quartic order Z[], order whose group of units is of rank equal to one. We partially prove this conjecture. This generalizes a result of T. Nagell, who proved in 1930 a similar result for real cubic units with two non real conjugates.
In this chapter we assume that the reader is acquainted with the ordinary ideal theory in number fields. Cf. for instance [B7]. The first two sections should be read as technical background for Chapter 10, §2. On the other hand, although we strive for some completeness, once the reader sees the first results that the proper o-lattices form a multiplicative group, he can wait to read the other results until he needs them, as they are slightly technical. They are all classical, known to Dedekind, except possibly for the fact that a proper o-lattice is locally principal, which seems to have been first pointed out by Ihara [26]. The localization technique will be used heavily for the idelic formulation of the complex multiplication, as in Shimura [B12].
The current paper considers the question of power bases in the cyclotomic number field Q(ζ), ζp = 1, p an odd prime. The ring of integers is Z[ζ], and there do exist further “non-obvious” generators for this order; specifically we shall see that Z[α] = Z[ζ] for . We conjecture that, up to conjugacy, there can be no further such integral generators, and prove that this is indeed the case in Q(ζ7).
Applying Baker’s effective method and the reduction procedure of Baker and Davenport, we present several lists of solutions of index form equations in (totally real and complex) cubic algebraic number fields. These solutions yield all power integral bases of these fields.
Let m, n be distinct square-free rational integers and let K=Q(√m, √n). Combining Baker-type inequalities with a suitable version of the Baker-Davenport reduction method we give a computational algorithm for determining all elements with minimal index in such number fields.
LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutionsp, qof that equation then yield all solutionsx, y, zof (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fieldsKcan be reduced to the resolution of a cubic equationF(u, v)=iand a corresponding system of quadratic equationsQ1(x, y, z)=u,Q2(x, y, z)=v, whereFis a binary cubic form andQ1, Q2are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic fieldK. The second application is to the calculation of all integral points of an elliptic curve.
Let K/ be a cyclic extension of degree l. Let ZK be the ring of integers of K. We say that ZK has a power basis (or is monogenic) if there exists θ ∈ ZK such that ZK = [θ]. We show that if l ≥ 5 is a prime, then ZK has no power basis, except in the well-known case where K is the maximal real subfield of a cyclotomic field; that is to say, if l is given, there exists, at most, one field K such that ZK has a power basis: the field , when 2l + 1 is prime (e.g., for l = 5, ZK has a power basis only for the field K = , and for l = 7, ZK never has a power basis).
Combining Baker's effective method with the reduction procedure of Baker and Davenport, we give an algorithm for the complete resolution of index form equations corresponding to totally real cyclic biquadratic number fields. The solutions make it possible to construct all power integral bases of these fields. The method can be modified to be applicable also to other types of decomposable form equations.
An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.
We give an efficient algorithm for the resolution of index form equations, especially for determining power integral bases, in sextic fields with an imaginary quadratic subfield. The method reduces the problem to the resolution of acubic relative Thue equationover the quadratic subfield. At the end of the paper we give a table containing the generators of all power integral bases in the first 25 fields of this type with smallest discriminant (in absolute value).1991 Mathematics Subject Classification:primary 11Y50, secondary 11Y40, 11D57
Einheitengleichungen in kommutativen Ringen
- G Niklasch
G. Niklasch, ``Einheitengleichungen in kommutativen Ringen,'' dissertation, Mathematisches Institut der Technischen Universita t Mu nchen, 1991.
Length: 45 pic 0 pts, 190 mm 9. I. Gaa l and M. Pohst, On the resolution of index form equations in sextic fields with an imaginary quadratic subfield
Codes: 4462 Signs: 1593. Length: 45 pic 0 pts, 190 mm
9. I. Gaa l and M. Pohst, On the resolution of index form equations in sextic fields with an
imaginary quadratic subfield, J. Symbolic Comput. 22 (1996), 4255434.
Indices in cyclic cubic fields
- Dummit