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On the Computation of the Norm-Euclidean Minimum Of International Journal of Algebra, Vol. 3, 2009, no. 7, 341 - 354 On the Computation of the Norm-Euclidean Minimum of Algebraic Number Fields Michele Elia Dipartimento di Elettronica Politecnico di Torino I-10129 Torino, Italy [email protected] J. Carmelo Interlando Department of Mathematics and Statistics San Diego State University, San Diego, CA, USA [email protected] Abstract Let F be an algebraic number field whose group of units has rank ≥ 1. The conjecture that the norm-Euclidean minimum M(F) is a rational number is affirmatively settled. It is proved that M(F) is lower bounded by the inverse of the smallest norm of all nonzero prime OF-ideals. Furthermore, when F/Q is a normal extension, the numer- ator and denominator of M(F) lie within finite sets of integers that can be explicitly calculated. As an application, it is proved that the known lower bounds of M(F) for 2 2 2 the cyclotomic field Q(ζ5) and the cyclic cubic fields of discriminants 103 , 109 , 117 , and 1572 are the actual values of M(F). Mathematics Subject Classification: 11R04, 13F07, 13A05, 11R27, 11Y40 Keywords: algebraic number fields, Euclidean rings, units, group actions I Introduction The problem of establishing whether an algebraic number field is a Euclidean domain has always attracted a lot of interest [6]. The Euclidean character of fields has been considered with respect to many different norms, with prevalence of the absolute value of the field norm. In this paper, F denotes an algebraic number field defined by a monic irreducible f x n Z x O F polynomial ( ) of degree in [ ]. As customary, denotes the ring of integers of . F O |N . | We say that is norm-Euclidean if its ring of integers is a Euclidean domain for ( ) , 342 M. Elia and J. Carmelo Interlando Q α, β ∈ O the absolute value of the field norm relative to . That is, given , there exist θ, ν ∈ O α θβ ν |N ν | < |N β | such that = + with ( ) ( ) . Most of the proofs concerning Euclidean domains are based on the following proposition, which is a consequence of the results contained in [5, Section 14.7, p. 212]: Proposition 1. The number field F is norm-Euclidean if and only if for every α ∈ F, β ∈ O |N α − β | < there exists such that ( ) 1. The norm-Euclidean minimum of F, denoted by M(F), is defined in [6] as M F |N ξ − ν | . ( ) = sup inf ( ) (1) ν∈Ç ξ∈ Note that F is norm-Euclidean or not according to whether M(F) < 1orM(F) > 1, respectively. If M(F) = 1, then both possibilities can occur, see [6]. In [12], van der Linden proved that M(F) is achievable if F has unit rank 1, that is, ξ F M F |N ξ0 | M F ∈ Q an element 0 in exists such that ( )= ( ) , which implies that ( ) .In [6], the Euclidean properties of number fields are surveyed and some surmises [6, p. 4] are put forth. In particular, it is conjectured that if F has unit rank ≥ 1 then: i) M(F) is achievable in F; a ii) M(F)= d where a, d ∈ Z. These conjectures are supported, or at least not contradicted, by Lemmermeyer’s extended tables [6] of number fields of given discriminant listed along with known or supposed norm- Euclidean minima. Norm-Euclidean quadratic fields have been completely classified,√ see [6, Section 4] for the succession of events that led to it. More specifically, Q( m) is norm-Euclidean if and only if m is one of the 21 integers: −11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73. In this paper, conjectures i) and ii) are affirmatively settled by extending an action F F/O Rn of the unit group of on to . That group action was considered by Barnes and Swinnerton-Dyer [6]. As a consequence, it is shown that M(F) is lower bounded by the O inverse of the smallest norm of all nonzero prime -ideals. Furthermore, it is shown that a in numerical evaluations of M(F)= d , the denominator d is taken from a finite set of integers which depends on the fundamental units of F. This set is explicitly defined for any F which is a normal extension of Q and has a group of units of rank greater than zero. Lastly, previously unknown values of M(F) for fields of small degree and discriminant are given in Table 1. Computation of the norm-Euclidean minimum 343 II Preliminaries Given an integral basis {ω1,...,ωn} for F,anyξ ∈ F can be represented either by a vector φ(ξ)inQn or by an n × n matrix D(ξ) with rational entries. The relation n ξ = xiωi,xi ∈ Q, i=1 n T n defines a bijective mapping φ : F → Q where for any ξ ∈ F, φ(ξ)=(x1,x2,...,xn) ∈ Q . φ F Qn φ O Zn F It follows that ( )= and ( )= . A representation of by matrices is defined by n D(ξ)= xiD(ωi),xi ∈ Q, i=1 (i) where D(ωi)=(djh) are integral matrices, see [14, pp. 1-3]. For i =1,...,n, the matrix D(ωi) is defined by the action of ωi on the row vector Ω = (ω1,...,ωn)as n (i) ωiωh = djhωj, 1 ≤ h ≤ n, j=1 that is, ωiΩ=ΩD(ωi). Note that D is a ring monomorphism from F into the ring M(n, Q) of n × n rational matrices. It follows, with some abuse of notation, that D(F) ⊆ M(n, Q) D O ⊆ M n, Z and ( ) ( ). Let σ1,...,σn be the embeddings of F in C, that is, the Q-isomorphisms of F in C. Then the eigenvalues of D(ξ) are ξ and its conjugates, that is, σ1(ξ),...,σn(ξ), since n n n ΩD(ξ)=Ω xiD(ωi)= xiΩD(ωi)= xiωiΩ=ξΩ , i=1 i=1 i=1 and σj(D(ξ)) = D(ξ) because the entries in D(ξ) are rational numbers. Using φ and D, the field product, trace, and norm are computed in F as: φ ξη D η φ ξ D ξ φ η , ξ D ξ , N ξ D ξ , ( )= ( ) ( )= ( ) ( ) Tr ( ) = tr( ( )) and ( ) = det( ( )) (2) respectively. n A complex number z of the form i=1 xiωi where the xi are rational or real numbers and at least one of the xi belongs to R \ Q will be called an irrational point of F. The representation φ(z)ofz is not unique, but for our purposes this fact is irrelevant. The irrational points of F do not belong to F, but they may be viewed as limit points of sequences of elements in F. The union of F and its irrational points will be denoted by F. Note that the representations φ and D can be both extended to F in the obvious way. Let Hn be the hypercube {(x1,...,xn) | 0 ≤ xi < 1} with vertex set Vn = {(x1,...,xn) | xi ∈ { , }} H F φ O Zn 0 1 . Note that n is the fundamental region n of the lattice ( )= . The image T φ F/O Hn n = ( ) is the set of points in with rational coordinates. 344 M. Elia and J. Carmelo Interlando ξ F ξ,O Definition 1. Given any point of , its distance dist( ) from the ring of integers O is defined as ξ,O |N ξ − ν | . dist( ) = inf ( ) ν∈Ç δ ∈ O ξ O δO Similarly, for any , the distance between and the integral -ideal is ξ,δO |N ξ − δν |. defined as dist( ) = infν∈Ç ( ) Although we will not be doing explicit calculations with irrational points, Definition 1 can be extended to irrational points z ∈ F with the assumption of referring to a specific representation φ(z). The distance of an z O irrational point to is then defined as the infimum of the distances over all possible representations of z. Definition 2. A rational or irrational point ξo ∈ F is said to be optimal if ξ , O ξ,O M F . dist( o ) = sup dist( )= ( ) n φ(ξ)∈Ê ς M F ς , O A rational or irrational optimal point o is called a 0-optimal point if ( ) = dist( o )= |N ςo | ( ) . Note that the search for the supreme can be restricted to points ξ ∈ F such that φ ξ ∈ Hn ξ,O O ς ∈ F o ( ) since dist( ) is defined modulo . A 0-optimal point can always |N · | R R+ be found since ( ) is a continuous function from into , the set of nonnegative real numbers, see [6, p. 7]. In other words, some rational or irrational optimal point ξo is always achievable. δ ∈ O Z δ O /δO δ For , let ( )= denote the residue class ring modulo [3, pp. 24-31]. The multiplicative subgroup of the elements prime with δ in Z(δ) is denoted by Z(δ)∗. The Z δ δ |Z δ | |N δ | order of ( ) is equal to the absolute value of the field norm of , that is, ( ) = ( ) . The order of Z(δ)∗ is equal to the Euler totient function ϕ(δ) counting the number of O /δO algebraic integers that are units in the ring , see [2, p. 231, Problem 4] for its ϕ δ properties. The function ( ) can be computed from the prime-power factorization of δ s βai O = i=1 i in . More specifically, s ϕ δ N δ − 1 , ( )= ( ) 1 (3) N βi i=1 ( ) see [2, p. 231, Problem 5] or [8, p. 152]. U F O Let ( ) be the group of units of .
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