Algebraic Number Field Of Degree Research Articles

FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $\sigma_1,\sigma_2,\dots,\sigma_d$ the embeddings of $K$ into $\mathbb{C}$, $\alpha$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-\sigma_i(\alpha) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-\sigma_i(\alpha\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effective upper bound for the solutions $(x,y,a)\in{\mathbb{Z}}^3$ of the Diophantine inequalities $$ 0<|F_a(x,y)|\le m $$ for which $xy\not=0$ and ${\mathbb{Q}}(\alpha \upsilon^a)=K$. Our estimate involves an effectively computable constant depending only on $d$; it is explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$, and in terms of the regulator of the number field $K$.

Some explicit zero-free regions for Hecke L-functions

Let K be an algebraic number field of degree n over Q and let dK denote the absolute value of its discriminant. Let χ be a Hecke character on K with conductor F(χ). We let L(s,χ) denote the Hecke L-function associated with χ. Set Aχ=dKNK/Q(F(χ)). In this paper we present some explicit zero-free regions for Hecke L-functions. For example we prove the following results using Stechkin's device: If K≠Q, the Dedekind zeta function ζK(s) has at most one zero ρ=β+iγ with β>1−(2log⁡dK)−1 and |γ|<(2log⁡dK)−1. This zero, if it exists, has to be real and simple; If χ is a primitive Hecke character on K of order 2, then L(s,χ) has at most one zero ρ=β+iγ with β>1−(4log⁡Aχ)−1 and |γ|<(4log⁡Aχ)−1. If such a zero exists, it has to be real and simple. Moreover, using approaches due to Heath-Brown and to Kadiri, we show that for a primitive Hecke character χ on K of order |χ|≥3, L(s,χ) has no zero ρ=β+iγ in the region β>1−(15.10log⁡Aχ)−1 and |γ|<13tan⁡(π8).

Computation of the Euclidean minimum of algebraic number fields

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri in 2007. With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 8 in any signature, especially for cyclotomic fields, and many new examples of norm-Euclidean or non-norm-Euclidean algebraic number fields. Then, we show how to apply the algorithm to study extensions of norm-Euclideanity.

Properties of transformation matrices between the normal bases in orders of cubic field

Abstract In this paper, the properties of transformation matrices between the normal bases in orders of algebraic number fields of degree 3 are shown.

Linear forms at a basis of an algebraic number field

It was proved by Cassels and Swinnerton-Dyer that the Littlewood conjecture in simultaneous Diophantine approximation holds for any pair of numbers in a cubic field. Later this result was generalized by Peck to a basis ( 1 , α 1 , … , α n ) of a real algebraic number field of degree at least 3. By transference, this result provides some solutions for the dual form of Littlewoodʼs conjecture. Here we find another solutions, and using Bakerʼs estimates for linear forms in logarithms of algebraic numbers, we discuss whether the result is best possible.

A Signature Scheme Based on the Intractability of Computing Roots

We present RDSA, a variant of the DSA signature scheme, whose security is based on the intractability of extracting roots in a finite abelian group. We prove that RDSA is secure against an adaptively chosen message attack in the random oracle model if and only if computing roots in the underlying group is intractable. We report on a very efficient implementation of RDSA in the class group of imaginary quadratic orders. We also show how to construct class groups of algebraic number fields of degree > 2 in which RDSA can be implemented.

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Let O denote the ring of integers of an algebraic number field of degree m which is totally and tamely ramified at the prime p. Write ζq = exp(2πi/q), where q = pr. We evaluate the twisted Kloosterman sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α)+z/N(α) q , where T and N denote trace and norm, and where χ is a Dirichlet character (mod q). This extends results of Salie for m = 1 and of Yangbo Ye for prime m dividing p− 1. Our method is based upon our evaluation of the Gauss sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α) q , which extends results of Mauclaire for m = 1.

The number of solutions of linear equations in roots of unity

Denote by ν(a1, . . . , an) the number of non-degenerate solutions of (1.1). First, let a1, . . . , an be non-zero rational numbers. In 1965, Mann [2] showed that if (ζ1, . . . , ζn) is a non-degenerate solution of (1.1), then ζ 1 = · · · = ζ n = 1, where d is a product of distinct primes ≤ n+1. From this result it can be deduced that ν(a1, . . . , an) ≤ e1 2 for some absolute constant c1. Later, Conway and Jones [1] showed that for every non-degenerate solution (ζ1, . . . , ζn) of (1.1) one has ζ 1 = · · · = ζ n = 1, where d is the product of distinct primes p1, . . . , pl with ∑l i=1(pi−2) ≤ n−1. This implies that ν(a1, . . . , an) ≤ e2 3/2(log n)1/2 for some absolute constant c2. Schinzel [3] showed that if a1, . . . , an are non-zero and generate an algebraic number field of degree D, then ν(a1, . . . , an) ≤ c2(n,D) for some function c2 depending only on n and D. Later, Zannier [5] gave a different proof of this fact and computed c2 explicitly. Finally, Schlickewei [4] succeeded to derive an upper bound for the number of non-degenerate solutions of (1.1) depending only on n for arbitrary complex coefficients a1, . . . , an.

The S5 Extensions of Degree 6 with Minimum Discriminant

The algebraic number fields of degree 6 having Galois group S5 and minimum discriminant are determined for signatures (0, 3), (2,2) and (6,0). The fields F0, F2, F6 are generated by roots of f0(t) = t6 3t4 + 2t3 + 6t2 + 1, f2(t) = t6 – 2t4 + 12t3 – 16t + 8, and f6(t) = t6 – 18t4 + 9t3 + 90t2 – 70t – 69 respectively. Each of these fields is unique up to isomorphism. This completes the enumeration of primitive sextic fields with min imum discriminant for all. possible combinations of Galois group and signature.

On a Conjecture of Shimura Concerning Periods of Hilbert Modular Forms

Introduction. In this paper, we shall give an affirmative answer to an essential part of the conjecture of Shimura on P-invariants of Hilbert modular forms. Let F be a totally real algebraic number field of degree n and JF be the set of all isomorphisms of F into C. Let FA (resp. FA) be the adele ring (resp. the idele group) of F and Fx be the archimedean part of FA. Let X be a primitive system of eigenvalues of Hecke operators which occurs in the space of holomorphic Hilbert modular cusp forms on GL(2, FA) of weight k and f be the primitive form which belongs to X. In [S 1], Shimura introduced an invariant u(e, f) E CX for every 6 E (Z/2Z)JF such that

Enumeration of Quartic Fields of Small Discriminant

With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant $< {10^6}$ have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The ${T_2}$ criterion of Pohst was applied to reduce the number of redundant examples.

Counting Algebraic Units with Bounded Height

Let K be an algebraic number field of degree d and let U denote its group of units. Suppose U has rank r and regulator R. Let U(q) denote the number of units in U with height less than q. We obtain an asymptotic formula for U(q) of the shape U(q) = A(log q)r + O((log q)r − 1 − δ), where δ is given explicitly in terms of d and R.

The Minimum Discriminant of Totally Real Algebraic Number Fields of Degree 9 with Cubic Subfields

The Minimum Discriminant of Totally Real Algebraic Number Fields of Degree 9 with Cubic Subfields

The minimum discriminant of totally real algebraic number fields of degree 9 with cubic subfields

With the help of the computer language UBASIC86, the minimum discriminant d ( K ) d(K) of totally real algebraic number fields K of degree 9 with cubic subfields F is determined. It is given by d ( K ) = 16240385609 d(K) = 16240385609 . The defining equation for K is given by f ( x ) = x 9 − x 8 − 9 x 7 + 4 x 6 + 26 x 5 − 2 x 4 − 25 x 3 − x 2 + 7 x + 1 f(x) = {x^9} - {x^8} - 9{x^7} + 4{x^6} + 26{x^5} - 2{x^4} - 25{x^3} - {x^2} + 7x + 1 , and K is uniquely determined by d ( K ) d(K) up to Q-isomorphism. The field K has the cubic subfield F with d ( F ) = 49 d(F) = 49 defined by the polynomial f ( x ) = x 3 + x 2 − 2 x − 1 f(x) = {x^3} + {x^2} - 2x - 1 .

Enumeration of quartic fields of small discriminant

With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant &gt; 10 6 &gt; {10^6} have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The T 2 {T_2} criterion of Pohst was applied to reduce the number of redundant examples.

Kronecker classes of field extensions of small degree

AbstractThe structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

A zero‐free region for the Hecke L ‐functions

MathematikaVolume 37, Issue 2 p. 287-304 Research Article A zero-free region for the Hecke L-functions M. D. Coleman, M. D. Coleman Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QDSearch for more papers by this author M. D. Coleman, M. D. Coleman Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QDSearch for more papers by this author First published: 01 December 1990 https://doi.org/10.1112/S0025579300013000Citations: 19AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume37, Issue2December 1990Pages 287-304 RelatedInformation

Exact bounds on the order of torsion points of curves of genus 1

Let k be an algebraic number field of degree n on ℚ2; ℱ and ℊ, respectively, the curves on k; let , and ℴm, ℴ'm be the bases of groups of all points of order m on ℱ and g, respectively. A proof of the following theorem is sketched: let p>3 be prime; if , then τ(pt)⩽6n; if k, then τ(pt)⩽4n. The resulting bounds are unimprovable.

Automorphy factors for a Hilbert modular group

Let K be a totally real algebraic number field of degree n &gt; 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

On unramified cyclic extensions of degree l of algebraic number fields of degree l

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.