Number Field Of Degree Research Articles

Ideal Class Groups, Some Computations

Starting from a base field with properties similar to those of the rational numbers, the structure of the ideal class group of a biquadratic dicyclic extension is examined. Class number relations and structural connections between the ideal class groups of the intermediate fields allow the determination of this structure in some cases. Explicit computations are performed for some number fields of degree 8.

Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤2

We determine all nonquadratic imaginary cyclic number fieldsKof 2-power degrees with ideal class groups of exponents≤2\leq 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some(Z/2Z)m,m≥0{({\mathbf {Z}}/2{\mathbf {Z}})^m},m \geq 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).

Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2

We determine all nonquadratic imaginary cyclic number fields K of 2-power degrees with ideal class groups of exponents ≤ 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some (Z/2Z) m , m ≥ 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11)

On Siegel zeros of Hecke-Landau zeta-functions

The primary concern of this paper is to deal with Siegel zeros of Hecke-Landau zeta-functions in an algebraic number field of finite degree over the rationals. As in the rational case with DirichletL-functions, the location of such zeros is closely connected with lower bounds for the corresponding zeta-functions at the points=1. This will be the theme in the first part of the paper. In this second part we first derive a form of the Brun-Titchmarsh theorem in the setting of a number field which is appropriate in our context. Then we turn our attention to the fact that an improvement of the constant in this inequality would lead to the nonexistence of Siegel zeros. The procedure is based on a weighted algebraic form of Selberg's upper bound sieve.

Sums and differences of quartic norms

MathematikaVolume 40, Issue 2 p. 233-245 Research Article Sums and differences of quartic norms Henryk Iwaniec, Henryk Iwaniec Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903 USASearch for more papers by this authorJacek Pomykala, Jacek Pomykala Institute of Mathematics, Warsaw University, 00-950 Warsaw, PolandSearch for more papers by this author Henryk Iwaniec, Henryk Iwaniec Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903 USASearch for more papers by this authorJacek Pomykala, Jacek Pomykala Institute of Mathematics, Warsaw University, 00-950 Warsaw, PolandSearch for more papers by this author First published: 01 December 1993 https://doi.org/10.1112/S0025579300007014Citations: 1AboutPDF 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 Citing Literature Volume40, Issue2December 1993Pages 233-245 RelatedInformation

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.

Witt equivalence classes of quartic number fields

It has recently been established that there are exactly seven Witt equivalence classes of quadratic number fields, and then all quadratic and cubic number fields have been classified with respect to Witt equivalence. In this paper we have classified number fields of degree four. Using this classification, we have proved the Conjecture of Szymiczek about the representability of Witt equivalence classes by quadratic extensions of quadratic fields.

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.

On the class number and unit index of simplest quartic fields

The term “simplest” field has been used to describe certain totally real, cyclic number fields of degrees 2, 3, 4, 5, 6, and 8. For each of these degrees, the fields are defined by a one-parameter family of polynomials with constant term ±1. The regulator of these “simplest” fields is small in an asymptotic sense: in consequence, the class number of these fields tends to be large.

A table of totally real quintic number fields

We give a table of the 1077 totally real number fields of degree five having a discriminant less than 2 000 000. There are two nonisomorphic fields of discriminant 1 810 969 and two nonisomorphic fields of discriminant 1 891 377. All the other number fields in the table are characterized by their discriminant. Among these fields, three are cyclic and four have a Galois closure whose Galois group is the dihedral group D 5 {D_5} . The Galois closure for all the other fields in the table has a Galois group isomorphic to S 5 {S_5} .

Leopoldt kernels and central extensions of algebraic number fields

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

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

Square-free values of cubic polynomials in algebraic number fields

Let K be an algebraic number field of finite degree over the rationals. The two themes of this paper are the problems about the existence of infinitely many power-free algebraic integers in polynomial sequences generated by irreducible polynomials of K with integer arguments and with prime arguments, respectively. In particular, it follows from the results that an irreducible cubic polynomial of K represents square-free numbers infinitely often, provided that an obvious necessary condition is fulfilled.

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.

Formulas for some diophantine approximation constants

AbstractLet M be a full Z-module in F a real number field of degree at least 3 with N(α) denoting the norm of α ∈ F. Given any nonzero number φ in M we make the plausible conjecture that one can find a number ß in M such that N(ß) = N(φ) and the algebraic conjugates of ß (not including ß) have ratios arbitraily near any given numbers consistent with the complex algebraic conjugates of elements of F. We use the conjecture to give explicit formulas for some diophantine approximation constants. Without the conjecture our methods lead to corresponding lower bounds for these constants.

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.