Last edited by Aragar
Tuesday, July 28, 2020 | History

2 edition of norm ideal bound for a class of biquadratic fields found in the catalog.

norm ideal bound for a class of biquadratic fields

L. J. Mordell

# norm ideal bound for a class of biquadratic fields

## by L. J. Mordell

Written in English

Subjects:
• Ideals (Algebra),
• Class field theory.

• Edition Notes

Classifications The Physical Object Statement by L. J. Mordell. Series Det kgl. Norske videnskabers selskab. Forhandlinger, bd. 42, 1969, nr. 9 LC Classifications AS283 .T82 bd. 42, 1969, nr. 9 Pagination 53-55 p. Number of Pages 55 Open Library OL5003333M LC Control Number 76513390

Math Class groups for imaginary quadratic fields In general it is a very diﬃcult problem to determine the class number of a number ﬁeld, let alone the structure of its class group. However, in the special case of imaginary quadratic ﬁelds there is a very explicit algorithm that determines the class . In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. This integral domain is a particular case of a commutative ring of quadratic does not have a total ordering that respects arithmetic.

Chapter 6 proves that ideals are uniformly distributed among classes of ideals. Concretely, the number of ideals in an arbitrary ideal class, of norm bounded by t>0, is asymptotically proportional to t, and the proportionality constant (independent of the chosen class) is determined. Further refinements of this are presented in the s: 7. istence of two primes that split completely in a real biquadratic field. For in-stance, the pri89 split completely in Q(Λ/2, \/5], hence Q ί y/—2 5 31 89 J has an infinite 2-class field tower; its 2-ideal class group is of type (4,2,2). Taussky-Todd [12] proved that a number field with 2-ideal class group of.

Fundamental unit system and class number of real bicyclic biquadratic number fields Wang, Kunpeng, Proceedings of the Japan Academy, Series A, Mathematical Sciences, ; Refined version of Hasse's Satz 45 on class number parity Ichimura, Humio, Tsukuba Journal of Mathematics, ; Inequalities of Ono numbers and class numbers associated to imaginary quadratic fields Shimizu, Kenichi. factor ideal $(3)$ in biquadratic field. Ask Question Asked 4 years, 5 months ago. Factorization of ideal in field $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure. 2. Splitting of a prime in a number field. 2. Computing the norm of a principal ideal in a quadratic number field. 3. Show every ideal in the maximal order of a quadratic.

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### Norm ideal bound for a class of biquadratic fields by L. J. Mordell Download PDF EPUB FB2

Get this from a library. A norm ideal bound for a class of biquadratic fields. [L J Mordell] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library.

Create. JOURNAL OF NUMBER THEORY I, () A Gauss Bound for a Class of Biquadratic Fields R. LAKEIN*t Department of Mathematics, University of Maryland, College Park, Maryland Communicated by Sigekatu Kuroda Received February 8, A bound is obtained for the minimal norm of the ideals in an ideal class of the number field K, where K is a quadratic extension of a Cited by: 3.

A gauss bound for a class of biquadratic fields. Abstract. A bound is obtained for the minimal norm of the ideals in an ideal class of the number field K, where K is a quadratic extension of a Euclidean imaginary quadratic field. Previous article in issue; Next article in issue;Cited by: 3.

A gauss bound for a class of biquadratic fields. By R.B. Lakein. Get PDF ( KB) AbstractA bound is obtained for the minimal norm of the ideals in an ideal class of the number field K, where K is a quadratic extension of a Euclidean imaginary quadratic field Publisher: Published by Elsevier Inc.

Year: Author: R.B. Lakein. Abstract. In this paper, we give an explicit lower bound for the class number of real quadratic field, where is a square-free integer, using which is the number of odd prime divisors of.

Introduction. Let be a positive square-free integer and let and denote the class number and the class group of a real quadratic field, respectively. The class number problem of quadratic fields is one Author: Hasan Sankari, Ahmad Issa.

H. Lenstra \\cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in norm ideal bound for a class of biquadratic fields book fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in \\cite{hester1} and \\cite{cathy}).

In this paper, we construct certain new families of biquadratic number fields having a non. Abstract. The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields $$K = k{\left({{\sqrt D }} \right)}$$ over k = F q (T).For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e.

g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG) 2 + cS, then h(D) ≥ degS. In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group J K /P K where J K is the group of fractional ideals of the ring of integers of K, and P K is its subgroup of principal class group is a measure of the extent to which unique factorization fails in the ring of integers of order of the group, which is finite, is called.

PDF | In this paper, we give an explicit lower bound for the class number of real quadratic field, where is a square-free integer, using which is the | Find, read and cite all the research you. A norm ideal bound for a class of biquadratic fields: Razmyšleniâ matematika: Reflections of a mathematician: Three lectures on Fermat's last theorem: Two papers on.

Author of Three lectures on Fermat's last theorem, A chapter in the theory of numbers, Diophantine equations, A norm ideal bound for a class of biquadratic fields, Reflections of a mathematician, Diophantine equations, Reflections of a mathematician.

Diophantine equations by L. J Mordell (Book) 22 editions published between and in 3 languages and held by WorldCat member libraries worldwide. [2] H. Cohn, A Classical Introduction to Algebraic Numbers and Class Fields, Springer Verlag2nd printing 1 [3] L.

Holzer, Zahlentheorie II, Teubner Verlag, Leipzig 1 [4] R. Lakein, A Gauss bound for a class of biquadratic number ﬁelds J. Number Theory 1 (), – 3. On the Imaginary Bicyclic Biquadratic Fields With Class-Number 2 By D. Buell, H. Williams and K. Williams* Abstract. Assuming that the list of imaginary quadratic number fields of class-number 4 is complete, a determination is made of all imaginary bicyclic biquadratic number fields of class.

Analytic lower bounds for relative class numbers and maximal real subfields of quaternion oetie CM-fields with ideal class groups of exponents 2. Here we show that under the assumption of a.

It has been proved by A. Baker [1] and H. Stark [7] that there exist exactly 9 imaginary quadratic fields of class-number one. On the other hand, G.F. Gauss has conjectured that there exist infinitely many real quadratic fields of class-number one, and the conjecture is now still unsolved.

An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.

The beginning of the book describes the basic theory of algebraic number fields, and the book finishes with class field theory.

The proofs use a small amount of group cohomology (you should be fine) and use the original, analytic method to prove the First (or Second depending on. Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share.

In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field Q with Galois group the Klein four-group. Structure and subfields. Biquadratic fields are all obtained by adjoining two square ore in explicit terms they have the form K = Q(√ a, √ b).

for rational numbers a and b. Then the field has class number one and an infinite unramified extension. Based on the theorem, with the aid of the PARI software, he found two such real biquadratic fields with prime discriminant, including the above example.

He also noted that one can prove similar results for the existence of such real quadratic fields, by finding a degree.cubic ﬁeld is 2a+3 – we extend their result up to norm a2. For X∈ Rlet Pa(X) be the number of primitive principal ideals I(in a given monogenic simplest cubic ﬁeld K) with norm N(I) ≤ X.

Recall that an ideal I(or an element β) is primitive if n∤I(or n∤β) for all n∈ Z≥2. Theorem Finite Fields 79 §1 Basic Properties of Finite Fields 79 Cubic and Biquadratic Reciprocity §1 The Ring Z[o>] §2 Residue Class Rings §3 Cubic Residue Character §1 The Norm of an Ideal §2 The Power Residue Symbol §3 The Stickelberger Relation