6 Ideal Norms and the Dedekind-Kummer Theorem
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MAXIMAL and NON-MAXIMAL ORDERS 1. Introduction Let K Be A
MAXIMAL AND NON-MAXIMAL ORDERS LUKE WOLCOTT Abstract. In this paper we compare and contrast various prop- erties of maximal and non-maximal orders in the ring of integers of a number field. 1. Introduction Let K be a number field, and let OK be its ring of integers. An order in K is a subring R ⊆ OK such that OK /R is finite as a quotient of abelian groups. From this definition, it’s clear that there is a unique maximal order, namely OK . There are many properties that all orders in K share, but there are also differences. The main difference between a non-maximal order and the maximal order is that OK is integrally closed, but every non-maximal order is not. The result is that many of the nice properties of Dedekind domains do not hold in arbitrary orders. First we describe properties common to both maximal and non-maximal orders. In doing so, some useful results and constructions given in class for OK are generalized to arbitrary orders. Then we de- scribe a few important differences between maximal and non-maximal orders, and give proofs or counterexamples. Several proofs covered in lecture or the text will not be reproduced here. Throughout, all rings are commutative with unity. 2. Common Properties Proposition 2.1. The ring of integers OK of a number field K is a Noetherian ring, a finitely generated Z-algebra, and a free abelian group of rank n = [K : Q]. Proof: In class we showed that OK is a free abelian group of rank n = [K : Q], and is a ring. -
Super-Multiplicativity of Ideal Norms in Number Fields
Super-multiplicativity of ideal norms in number fields Stefano Marseglia Abstract In this article we study inequalities of ideal norms. We prove that in a subring R of a number field every ideal can be generated by at most 3 elements if and only if the ideal norm satisfies N(IJ) ≥ N(I)N(J) for every pair of non-zero ideals I and J of every ring extension of R contained in the normalization R˜. 1 Introduction When we are studying a number ring R, that is a subring of a number field K, it can be useful to understand the size of its ideals compared to the whole ring. The main tool for this purpose is the norm map which associates to every non-zero ideal I of R its index as an abelian subgroup N(I) = [R : I]. If R is the maximal order, or ring of integers, of K then this map is multiplicative, that is, for every pair of non-zero ideals I,J ⊆ R we have N(I)N(J) = N(IJ). If the number ring is not the maximal order this equality may not hold for some pair of non-zero ideals. For example, arXiv:1810.02238v2 [math.NT] 1 Apr 2020 if we consider the quadratic order Z[2i] and the ideal I = (2, 2i), then we have that N(I)=2 and N(I2)=8, so we have the inequality N(I2) > N(I)2. Observe that if every maximal ideal p of a number ring R satisfies N(p2) ≤ N(p)2, then we can conclude that R is the maximal order of K (see Corollary 2.8). -
Class Group Computations in Number Fields and Applications to Cryptology Alexandre Gélin
Class group computations in number fields and applications to cryptology Alexandre Gélin To cite this version: Alexandre Gélin. Class group computations in number fields and applications to cryptology. Data Structures and Algorithms [cs.DS]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066398. tel-01696470v2 HAL Id: tel-01696470 https://tel.archives-ouvertes.fr/tel-01696470v2 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par Alexandre GÉLIN Pour obtenir le grade de DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE Calcul de Groupes de Classes d’un Corps de Nombres et Applications à la Cryptologie Thèse dirigée par Antoine JOUX et Arjen LENSTRA soutenue le vendredi 22 septembre 2017 après avis des rapporteurs : M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern devant le jury composé de : M. Karim BELABAS Professeur, Université de Bordeaux M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern M. -
DISCRIMINANTS in TOWERS Let a Be a Dedekind Domain with Fraction
DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K=F be a finite separable ex- tension field, and let B be the integral closure of A in K. In this note, we will define the discriminant ideal B=A and the relative ideal norm NB=A(b). The goal is to prove the formula D [L:K] C=A = NB=A C=B B=A , D D ·D where C is the integral closure of B in a finite separable extension field L=K. See Theo- rem 6.1. The main tool we will use is localizations, and in some sense the main purpose of this note is to demonstrate the utility of localizations in algebraic number theory via the discriminants in towers formula. Our treatment is self-contained in that it only uses results from Samuel’s Algebraic Theory of Numbers, cited as [Samuel]. Remark. All finite extensions of a perfect field are separable, so one can replace “Let K=F be a separable extension” by “suppose F is perfect” here and throughout. Note that Samuel generally assumes the base has characteristic zero when it suffices to assume that an extension is separable. We will use the more general fact, while quoting [Samuel] for the proof. 1. Notation and review. Here we fix some notations and recall some facts proved in [Samuel]. Let K=F be a finite field extension of degree n, and let x1,..., xn K. We define 2 n D x1,..., xn det TrK=F xi x j . -
Super-Multiplicativity of Ideal Norms in Number Fields
Universiteit Leiden Mathematisch Instituut Master Thesis Super-multiplicativity of ideal norms in number fields Academic year 2012-2013 Candidate: Advisor: Stefano Marseglia Prof. Bart de Smit Contents 1 Preliminaries 1 2 Quadratic and quartic case 10 3 Main theorem: first implication 14 4 Main theorem: second implication 19 Introduction When we are studying a number ring R, that is a subring of a number field K, it can be useful to understand \how big" its ideals are compared to the whole ring. The main tool for this purpose is the norm map: N : I(R) −! Z>0 I 7−! #R=I where I(R) is the set of non-zero ideals of R. It is well known that this map is multiplicative if R is the maximal order, or ring of integers of the number field. This means that for every pair of ideals I;J ⊆ R we have: N(I)N(J) = N(IJ): For an arbitrary number ring in general this equality fails. For example, if we consider the quadratic order Z[2i] and the ideal I = (2; 2i), then we have that N(I) = 2 and N(I2) = 8, so we have the inequality N(I2) > N(I)2. In the first chapter we will recall some theorems and useful techniques of commutative algebra and algebraic number theory that will help us to understand the behaviour of the ideal norm. In chapter 2 we will see that the inequality of the previous example is not a coincidence. More precisely we will prove that in any quadratic order, for every pair of ideals I;J we have that N(IJ) ≥ N(I)N(J). -
Dirichlet's Class Number Formula
Dirichlet's Class Number Formula Luke Giberson 4/26/13 These lecture notes are a condensed version of Tom Weston's exposition on this topic. The goal is to develop an analytic formula for the class number of an imaginary quadratic field. Algebraic Motivation Definition.p Fix a squarefree positive integer n. The ring of algebraic integers, O−n ≤ Q( −n) is defined as ( p a + b −n a; b 2 Z if n ≡ 1; 2 mod 4 O−n = p a + b −n 2a; 2b 2 Z; 2a ≡ 2b mod 2 if n ≡ 3 mod 4 or equivalently O−n = fa + b! : a; b 2 Zg; where 8p −n if n ≡ 1; 2 mod 4 < p ! = 1 + −n : if n ≡ 3 mod 4: 2 This will be our primary object of study. Though the conditions in this def- inition appear arbitrary,p one can check that elements in O−n are precisely those elements in Q( −n) whose characteristic polynomial has integer coefficients. Definition. We define the norm, a function from the elements of any quadratic field to the integers as N(α) = αα¯. Working in an imaginary quadratic field, the norm is never negative. The norm is particularly useful in transferring multiplicative questions from O−n to the integers. Here are a few properties that are immediate from the definitions. • If α divides β in O−n, then N(α) divides N(β) in Z. • An element α 2 O−n is a unit if and only if N(α) = 1. • If N(α) is prime, then α is irreducible in O−n . -
Multiplication Semimodules
Acta Univ. Sapientiae, Mathematica, 11, 1 (2019) 172{185 DOI: 10.2478/ausm-2019-0014 Multiplication semimodules Rafieh Razavi Nazari Shaban Ghalandarzadeh Faculty of Mathematics, Faculty of Mathematics, K. N. Toosi University of Technology, K. N. Toosi University of Technology, Tehran, Iran Tehran, Iran email: [email protected] email: [email protected] Abstract. Let S be a semiring. An S-semimodule M is called a mul- tiplication semimodule if for each subsemimodule N of M there exists an ideal I of S such that N = IM. In this paper we investigate some properties of multiplication semimodules and generalize some results on multiplication modules to semimodules. We show that every multiplica- tively cancellative multiplication semimodule is finitely generated and projective. Moreover, we characterize finitely generated cancellative mul- tiplication S-semimodules when S is a yoked semiring such that every maximal ideal of S is subtractive. 1 Introduction In this paper, we study multiplication semimodules and extend some results of [7] and [17] to semimodules over semirings. A semiring is a nonempty set S together with two binary operations addition (+) and multiplication (·) such that (S; +) is a commutative monoid with identity element 0; (S; :) is a monoid with identity element 1 6= 0; 0a = 0 = a0 for all a 2 S; a(b + c) = ab + ac and (b + c)a = ba + ca for every a; b; c 2 S. We say that S is a commutative semiring if the monoid (S; :) is commutative. In this paper we assume that all semirings are commutative. A nonempty subset I of a semiring S is called an ideal of S if a + b 2 I and sa 2 I for all a; b 2 I and s 2 S. -
NORM GROUPS with TAME RAMIFICATION 11 Is a Homomorphism, Which Is Equivalent to the first Equation of Bimultiplicativity (The Other Follows by Symmetry)
LECTURE 3 Norm Groups with Tame Ramification Let K be a field with char(K) 6= 2. Then × × 2 K =(K ) ' fcontinuous homomorphisms Gal(K) ! Z=2Zg ' fdegree 2 étale algebras over Kg which is dual to our original statement in Claim 1.8 (this result is a baby instance of Kummer theory). Npo te that an étale algebra over K is either K × K or a quadratic extension K( d)=K; the former corresponds to the trivial coset of squares in K×=(K×)2, and the latter to the coset defined by d 2 K× . If K is local, then lcft says that Galab(K) ' Kd× canonically. Combined (as such homomorphisms certainly factor through Galab(K)), we obtain that K×=(K×)2 is finite and canonically self-dual. This is equivalent to asserting that there exists a “suÿciently nice” pairing (·; ·): K×=(K×)2 × K×=(K×)2 ! f1; −1g; that is, one which is bimultiplicative, satisfying (a; bc) = (a; b)(a; c); (ab; c) = (a; c)(b; c); and non-degenerate, satisfying the condition if (a; b) = 1 for all b; then a 2 (K×)2 : We were able to give an easy definition of this pairing, namely, (a; b) = 1 () ax2 + by2 = 1 has a solution in K: Note that it is clear from this definition that (a; b) = (b; a), but unfortunately neither bimultiplicativity nor non-degeneracy is obvious, though we will prove that they hold in this lecture in many cases. We have shown in Claim 2.11 that a less symmetric definition of the Hilbert symbol holds, namely that for all a, p (a; b) = 1 () b is a norm in K( a)=K = K[t]=(t2 − a); which if a is a square, is simply isomorphic to K × K and everything is a norm. -
Computing the Power Residue Symbol
Radboud University Master Thesis Computing the power residue symbol Koen de Boer supervised by dr. W. Bosma dr. H.W. Lenstra Jr. August 28, 2016 ii Foreword Introduction In this thesis, an algorithm is proposed to compute the power residue symbol α b m in arbitrary number rings containing a primitive m-th root of unity. The algorithm consists of three parts: principalization, reduction and evaluation, where the reduction part is optional. The evaluation part is a probabilistic algorithm of which the expected running time might be polynomially bounded by the input size, a presumption made plausible by prime density results from analytic number theory and timing experiments. The principalization part is also probabilistic, but it is not tested in this thesis. The reduction algorithm is deterministic, but might not be a polynomial- time algorithm in its present form. Despite the fact that this reduction part is apparently not effective, it speeds up the overall process significantly in practice, which is the reason why it is incorporated in the main algorithm. When I started writing this thesis, I only had the reduction algorithm; the two other parts, principalization and evaluation, were invented much later. This is the main reason why this thesis concentrates primarily on the reduction al- gorithm by covering subjects like lattices and lattice reduction. Results about the density of prime numbers and other topics from analytic number theory, on which the presumed effectiveness of the principalization and evaluation al- gorithm is based, are not as extensively treated as I would have liked to. Since, in the beginning, I only had the reduction algorithm, I tried hard to prove that its running time is polynomially bounded. -
Mension One in Noetherian Rings R
LECTURE 5-12 Continuing in Chapter 11 of Eisenbud, we prove further results about ideals of codi- mension one in Noetherian rings R. Say that an ideal I has pure codimension one if every associated prime ideal of I (that is, of the quotient ring R=I as an R-module) has codimen- sion one; the term unmixed is often used instead of pure. The trivial case I = R is included in the definition. Then if R is a Noetherian domain such that for every maximal ideal P the ring RP is a UFD and if I ⊂ R is an ideal, then I is invertible (as a module or an ideal) if and only if it has pure codimension one. If I ⊂ K(R) is an invertible fractional ideal, then I is uniquely expressible as a finite product of powers of prime ideals of codimension one. Thus C(R) is a free abelian group generated by the codimension one primes of R. To prove this suppose first that I is invertible. If we localize at any maximal ideal then I becomes principal, generated by a non-zero-divisor. Since a UFD is integrally closed, an earlier theorem shows that I is pure of codimension one. Conversely, if P is a prime ideal of codimension one and M is maximal, then either P ⊂ M, in which case PM ⊂ RM is ∼ principal since RM is a UFD, or P 6⊂ M, in which case PM = RM ; in both case PM = RM , so P is invertible. Now we show that any ideal I of pure codimension one is a finite product of codi- mension one primes. -
Electromagnetic Time Reversal Electromagnetic Time Reversal
Electromagnetic Time Reversal Electromagnetic Time Reversal Application to Electromagnetic Compatibility and Power Systems Edited by Farhad Rachidi Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland Marcos Rubinstein University of Applied Sciences of Western Switzerland, Yverdon, Switzerland Mario Paolone Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland This edition first published 2017 © 2017 John Wiley & Sons, Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Farhad Rachidi, Marcos Rubinstein, and Mario Paolone to be identified asthe authors of the editorial material in this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty The publisher and the authors make no representations or warranties with respect tothe accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. -
Low-Dimensional Algebraic K-Theory of Dedekind Domains
Low-dimensional algebraic K-theory of Dedekind domains What is K-theory? k0¹OFº Z ⊕ Cl¹OFº Relationship with topological K-theory illen was the first to give appropriate definitions of algebraic K-theory. We will be This remarkable fact relates the K0 group of a Dedekind domain to its class Topological K-theory was developed before its algebraic counterpart by Atiyah and using his Q-construction because it is defined more generally, for any exact category C: group, which measures the failure of unique prime factorization. We follow the Hirzebruch. The K0 group was originally just denoted K, and it was defined for a proof outlined in »2; 1:1¼: compact Hausdor space X as: Kn¹Cº := πn+1¹BQCº 1. It is easy to show that every finitely generated projective OF-module is a K¹Xº := K¹VectF¹Xºº We will be focusing on a special exact category, the category of finitely generated direct sum of O -ideals. ¹ º F ¹ º projective modules over a ring R. We denote this by P R . By abuse of notation, we 2. Conversely, every fractional O -ideal I is a finitely generated projective Where Vectf X is the exact category of isomorphism classes of finite-dimensional F R C define the K groups of a ring R to be: OF-module, since: F-vector bundles on X under Whitney sum. Here F is either or . Kn¹Rº := Kn¹P¹Rºº 2.1 By a clever use of Chinese remainder theorem, any fractional ideal of a Dedekind Then a theorem of Swan gives an explicit relationship between this topological domain can be generated by at most 2 elements.