Computing the Power Residue Symbol
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The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
Number Theoretic Symbols in K-Theory and Motivic Homotopy Theory
Number Theoretic Symbols in K-theory and Motivic Homotopy Theory Håkon Kolderup Master’s Thesis, Spring 2016 Abstract We start out by reviewing the theory of symbols over number fields, emphasizing how this notion relates to classical reciprocity lawsp and algebraic pK-theory. Then we compute the second algebraic K-group of the fields pQ( −1) and Q( −3) based on Tate’s technique for K2(Q), and relate the result for Q( −1) to the law of biquadratic reciprocity. We then move into the realm of motivic homotopy theory, aiming to explain how symbols in number theory and relations in K-theory and Witt theory can be described as certain operations in stable motivic homotopy theory. We discuss Hu and Kriz’ proof of the fact that the Steinberg relation holds in the ring π∗α1 of stable motivic homotopy groups of the sphere spectrum 1. Based on this result, Morel identified the ring π∗α1 as MW the Milnor-Witt K-theory K∗ (F ) of the ground field F . Our last aim is to compute this ring in a few basic examples. i Contents Introduction iii 1 Results from Algebraic Number Theory 1 1.1 Reciprocity laws . 1 1.2 Preliminary results on quadratic fields . 4 1.3 The Gaussian integers . 6 1.3.1 Local structure . 8 1.4 The Eisenstein integers . 9 1.5 Class field theory . 11 1.5.1 On the higher unit groups . 12 1.5.2 Frobenius . 13 1.5.3 Local and global class field theory . 14 1.6 Symbols over number fields . -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
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). -
On the Class Group and the Local Class Group of a Pullback
JOURNAL OF ALGEBRA 181, 803]835Ž. 1996 ARTICLE NO. 0147 On the Class Group and the Local Class Group of a Pullback Marco FontanaU Dipartimento di Matematica, Uni¨ersitaÁ degli Studi di Roma Tre, Rome, Italy View metadata, citation and similar papers at core.ac.ukand brought to you by CORE provided by Elsevier - Publisher Connector Stefania GabelliU Dipartimento di Matematica, Uni¨ersitaÁ di Roma ``La Sapienza'', Rome, Italy Communicated by Richard G. Swan Received October 3, 1994 0. INTRODUCTION AND PRELIMINARY RESULTS Let M be a nonzero maximal ideal of an integral domain T, k the residue field TrM, w: T ª k the natural homomorphism, and D a proper subring of k. In this paper, we are interested in deepening the study of the Ž.fractional ideals and of the class group of the integral domain R arising from the following pullback of canonical homomorphisms: R [ wy1 Ž.DD6 6 6 Ž.I w6 TksTrM More precisely, we compare the Picard group, the class group, and the local class group of R with those of the integral domains D and T.We recall that as inwx Bo2 the class group of an integral domain A, CŽ.A ,is the group of t-invertibleŽ. fractional t-ideals of A modulo the principal ideals. The class group contains the Picard group, PicŽ.A , and, as inwx Bo1 , U Partially supported by the research funds of Ministero dell'Uni¨ersitaÁ e della Ricerca Scientifica e Tecnologica and by Grant 9300856.CT01 of Consiglio Nazionale delle Ricerche. 803 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. -
Gsm073-Endmatter.Pdf
http://dx.doi.org/10.1090/gsm/073 Graduat e Algebra : Commutativ e Vie w This page intentionally left blank Graduat e Algebra : Commutativ e View Louis Halle Rowen Graduate Studies in Mathematics Volum e 73 KHSS^ K l|y|^| America n Mathematica l Societ y iSyiiU ^ Providence , Rhod e Islan d Contents Introduction xi List of symbols xv Chapter 0. Introduction and Prerequisites 1 Groups 2 Rings 6 Polynomials 9 Structure theories 12 Vector spaces and linear algebra 13 Bilinear forms and inner products 15 Appendix 0A: Quadratic Forms 18 Appendix OB: Ordered Monoids 23 Exercises - Chapter 0 25 Appendix 0A 28 Appendix OB 31 Part I. Modules Chapter 1. Introduction to Modules and their Structure Theory 35 Maps of modules 38 The lattice of submodules of a module 42 Appendix 1A: Categories 44 VI Contents Chapter 2. Finitely Generated Modules 51 Cyclic modules 51 Generating sets 52 Direct sums of two modules 53 The direct sum of any set of modules 54 Bases and free modules 56 Matrices over commutative rings 58 Torsion 61 The structure of finitely generated modules over a PID 62 The theory of a single linear transformation 71 Application to Abelian groups 77 Appendix 2A: Arithmetic Lattices 77 Chapter 3. Simple Modules and Composition Series 81 Simple modules 81 Composition series 82 A group-theoretic version of composition series 87 Exercises — Part I 89 Chapter 1 89 Appendix 1A 90 Chapter 2 94 Chapter 3 96 Part II. AfRne Algebras and Noetherian Rings Introduction to Part II 99 Chapter 4. Galois Theory of Fields 101 Field extensions 102 Adjoining -
Identifying the Matrix Ring: Algorithms for Quaternion Algebras And
IDENTIFYING THE MATRIX RING: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS JOHN VOIGHT Abstract. We discuss the relationship between quaternion algebras and qua- dratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders. Contents 1. Rings and algebras 3 Computable rings and algebras 3 Quaternion algebras 5 Modules over Dedekind domains 6 2. Standard involutions and degree 6 Degree 7 Standard involutions 7 Quaternion orders 8 Algorithmically identifying a standard involution 9 3. Algebras with a standard involution and quadratic forms 10 Quadratic forms over rings 10 Quadratic forms over DVRs 11 Identifying quaternion algebras 14 Identifying quaternion orders 15 4. Identifying the matrix ring 16 Zerodivisors 16 arXiv:1004.0994v2 [math.NT] 30 Apr 2012 Conics 17 5. Splitting fields and the Hilbert symbol 19 Hilbert symbol 19 Local Hilbert symbol 20 Representing local fields 22 Computing the local Hilbert symbol 22 6. The even local Hilbert symbol 23 Computing the Jacobi symbol 26 Relationship to conics 28 Date: May 28, 2018. 1991 Mathematics Subject Classification. Primary 11R52; Secondary 11E12. Key words and phrases. Quadratic forms, quaternion algebras, maximal orders, algorithms, matrix ring, number theory. 1 2 JOHN VOIGHT 7. Maximal orders 28 Computing maximal orders, generally 28 Discriminants 29 Computing maximal orders 29 Complexity analysis 32 Relationship to conics 34 8. -
6 Ideal Norms and the Dedekind-Kummer Theorem
18.785 Number theory I Fall 2017 Lecture #6 09/25/2017 6 Ideal norms and the Dedekind-Kummer theorem In order to better understand how ideals split in Dedekind extensions we want to extend our definition of the norm map to ideals. Recall that for a ring extension B=A in which B is a free A-module of finite rank, we defined the norm map NB=A : B ! A as ×b NB=A(b) := det(B −! B); the determinant of the multiplication-by-b map with respect to an A-basis for B. If B is a free A-module we could define the norm of a B-ideal to be the A-ideal generated by the norms of its elements, but in the case we are most interested in (our \AKLB" setup) B is typically not a free A-module (even though it is finitely generated as an A-module). To get around this limitation, we introduce the notion of the module index, which we will use to define the norm of an ideal. In the special case where B is a free A-module, the norm of a B-ideal will be equal to the A-ideal generated by the norms of elements. 6.1 The module index Our strategy is to define the norm of a B-ideal as the intersection of the norms of its localizations at maximal ideals of A (note that B is an A-module, so we can view any ideal of B as an A-module). Recall that by Proposition 2.6 any A-module M in a K-vector space is equal to the intersection of its localizations at primes of A; this applies, in particular, to ideals (and fractional ideals) of A and B. -
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). -
Casting Retrospective Glances of David Hilbert in the History of Pure Mathematics
British Journal of Science 157 February 2012, Vol. 4 (1) Casting Retrospective Glances of David Hilbert in the History of Pure Mathematics W. Obeng-Denteh and S.K. Manu Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Abstract This study was conducted with the view of reconfirming the historical contributions of David Hilbert in the form of a review in the area of mathematics. It emerged that he contributed to the growth of especially the Nullstellensatz and was the first person to prove it. The implications and conclusions are that the current trend in the said area of mathematics could be appreciated owing to contributions and motivations of such a mathematician. Keywords: David Hilbert, Nullstellensatz, affine variety, natural number, general contribution 1.0 Introduction The term history came from the Greek word ἱστορία - historia, which means "inquiry, knowledge acquired by investigation‖( Santayana, …….) and it is the study of the human past. History can also mean the period of time after writing was invented. David Hilbert was a German mathematician who was born on 23rd January, 1862 and his demise occurred on 14th February, 1943. His parents were Otto and Maria Therese. He was born in the Province of Prussia, Wahlau and was the first of two children and only son. As far as the 19th and early 20th centuries are concerned he is deemed as one of the most influential and universal mathematicians. He reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundations of mathematics. His work in 1909 on integral equations led to 20th-century research in functional analysis. -
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 .