Algebraic Number Theory — Lecture Notes
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Multiplicative Theory of Ideals This Is Volume 43 in PURE and APPLIED MATHEMATICS a Series of Monographs and Textbooks Editors: PAULA
Multiplicative Theory of Ideals This is Volume 43 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume MULTIPLICATIVE THEORY OF IDEALS MAX D. LARSEN / PAUL J. McCARTHY University of Nebraska University of Kansas Lincoln, Nebraska Lawrence, Kansas @ A CADEM I C P RE S S New York and London 1971 COPYRIGHT 0 1971, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London WlX 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-137621 AMS (MOS)1970 Subject Classification 13F05; 13A05,13B20, 13C15,13E05,13F20 PRINTED IN THE UNITED STATES OF AMERICA To Lillie and Jean This Page Intentionally Left Blank Contents Preface xi ... Prerequisites Xlll Chapter I. Modules 1 Rings and Modules 1 2 Chain Conditions 8 3 Direct Sums 12 4 Tensor Products 15 5 Flat Modules 21 Exercises 27 Chapter II. Primary Decompositions and Noetherian Rings 1 Operations on Ideals and Submodules 36 2 Primary Submodules 39 3 Noetherian Rings 44 4 Uniqueness Results for Primary Decompositions 48 Exercises 52 Chapter Ill. Rings and Modules of Quotients 1 Definition 61 2 Extension and Contraction of Ideals 66 3 Properties of Rings of Quotients 71 Exercises 74 Vii Vlll CONTENTS Chapter IV. -
Exercises and Solutions in Groups Rings and Fields
EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets. -
Open Problems
Open Problems 1. Does the lower radical construction for `-rings stop at the first infinite ordinal (Theorems 3.2.5 and 3.2.8)? This is the case for rings; see [ADS]. 2. If P is a radical class of `-rings and A is an `-ideal of R, is P(A) an `-ideal of R (Theorem 3.2.16 and Exercises 3.2.24 and 3.2.25)? 3. (Diem [DI]) Is an sp-`-prime `-ring an `-domain (Theorem 3.7.3)? 4. If R is a torsion-free `-prime `-algebra over the totally ordered domain C and p(x) 2C[x]nC with p0(1) > 0 in C and p(a) ¸ 0 for every a 2 R, is R an `-domain (Theorems 3.8.3 and 3.8.4)? 5. Does each f -algebra over a commutative directed po-unital po-ring C whose left and right annihilator ideals vanish have a unique tight C-unital cover (Theorems 3.4.5, 3.4.11, and 3.4.12 and Exercises 3.4.16 and 3.4.18)? 6. Can an f -ring contain a principal `-idempotent `-ideal that is not generated by an upperpotent element (Theorem 3.4.4 and Exercise 3.4.15)? 7. Is each C-unital cover of a C-unitable (totally ordered) f -algebra (totally or- dered) tight (Theorem 3.4.10)? 8. Is an archimedean `-group a nonsingular module over its f -ring of f -endomorphisms (Exercise 4.3.56)? 9. If an archimedean `-group is a finitely generated module over its ring of f -endomorphisms, is it a cyclic module (Exercise 3.6.20)? 10. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
1. Rings of Fractions
1. Rings of Fractions 1.0. Rings and algebras (1.0.1) All the rings considered in this work possess a unit element; all the modules over such a ring are assumed unital; homomorphisms of rings are assumed to take unit element to unit element; and unless explicitly mentioned otherwise, all subrings of a ring A are assumed to contain the unit element of A. We generally consider commutative rings, and when we say ring without specifying, we understand this to mean a commutative ring. If A is a not necessarily commutative ring, we consider every A module to be a left A- module, unless we expressly say otherwise. (1.0.2) Let A and B be not necessarily commutative rings, φ : A ! B a homomorphism. Every left (respectively right) B-module M inherits the structure of a left (resp. right) A-module by the formula a:m = φ(a):m (resp m:a = m.φ(a)). When it is necessary to distinguish the A-module structure from the B-module structure on M, we use M[φ] to denote the left (resp. right) A-module so defined. If L is an A module, a homomorphism u : L ! M[φ] is therefore a homomorphism of commutative groups such that u(a:x) = φ(a):u(x) for a 2 A, x 2 L; one also calls this a φ-homomorphism L ! M, and the pair (u,φ) (or, by abuse of language, u) is a di-homomorphism from (A, L) to (B, M). The pair (A,L) made up of a ring A and an A module L therefore form the objects of a category where the morphisms are di-homomorphisms. -
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. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
Quantizations of Regular Functions on Nilpotent Orbits 3
QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS IVAN LOSEV Abstract. We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa. 1. Introduction 1.1. Nilpotent orbits and their quantizations. Let G be a connected semisimple algebraic group over C and let g be its Lie algebra. Pick a nilpotent orbit O ⊂ g. This orbit is a symplectic algebraic variety with respect to the Kirillov-Kostant form. So the algebra C[O] of regular functions on O acquires a Poisson bracket. This algebra is also naturally graded and the Poisson bracket has degree −1. So one can ask about quantizations of O, i.e., filtered algebras A equipped with an isomorphism gr A −→∼ C[O] of graded Poisson algebras. We are actually interested in quantizations that have some additional structures mir- roring those of O. Namely, the group G acts on O and the inclusion O ֒→ g is a moment map for this action. We want the G-action on C[O] to lift to a filtration preserving action on A. -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
Algebraic Numbers
FORMALIZED MATHEMATICS Vol. 24, No. 4, Pages 291–299, 2016 DOI: 10.1515/forma-2016-0025 degruyter.com/view/j/forma Algebraic Numbers Yasushige Watase Suginami-ku Matsunoki 6 3-21 Tokyo, Japan Summary. This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field Q induced by substitution of an algebraic number to the polynomial ring of Q[x] turns to be a field. MSC: 11R04 13B21 03B35 Keywords: algebraic number; integral dependency MML identifier: ALGNUM 1, version: 8.1.05 5.39.1282 1. Preliminaries From now on i, j denote natural numbers and A, B denote rings. Now we state the propositions: (1) Let us consider rings L1, L2, L3. Suppose L1 is a subring of L2 and L2 is a subring of L3. Then L1 is a subring of L3. (2) FQ is a subfield of CF. (3) FQ is a subring of CF. R (4) Z is a subring of CF. Let us consider elements x, y of B and elements x1, y1 of A. Now we state the propositions: (5) If A is a subring of B and x = x1 and y = y1, then x + y = x1 + y1. (6) If A is a subring of B and x = x1 and y = y1, then x · y = x1 · y1. -
7902127 Fordt DAVID JAKES Oni the COMPUTATION of THE
7902127 FORDt DAVID JAKES ON i THE COMPUTATION OF THE MAXIMAL ORDER IK A DEDEKIND DOMAIN. THE OHIO STATE UNIVERSITY, PH.D*, 1978 University. Microfilms International 3q o n . z e e b r o a o . a n n a r b o r , mi 48ios ON THE COMPUTATION OF THE MAXIMAL ORDER IN A DEDEKIND DOMAIN DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By David James Ford, B.S., M.S. # * * K * The Ohio State University 1978 Reading Committee: Approved By Paul Ponomarev Jerome Rothstein Hans Zassenhaus Hans Zassenhaus Department of Mathematics ACKNOWLDEGMENTS I must first acknowledge the help of my Adviser, Professor Hans Zassenhaus, who was a fountain of inspiration whenever I was dry. I must also acknowlegde the help, financial and otherwise, that my parents gave me. Without it I could not have finished this work. All of the computing in this project, amounting to between one and two thousand hours of computer time, was done on a PDP-11 belonging to Drake and Ford Engineers, through the generosity of my father. VITA October 28, 1946.......... Born - Columbus, Ohio 1967...................... B.S. in Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 1967-196 8 ................. Teaching Assistant Department of Mathematics The Ohio State University Columbus, Ohio 1968-197 1................. Programmer Children's Hospital Medical Center Boston, Massachusetts 1971-1975................. Teaching Associate Department of Mathematics The Ohio State University Columbus, Ohio 1975-1978................. Programmer Prindle and Patrick, Architects Columbus, Ohio PUBLICATIONS "Relations in Q (Zh X ZQ) and Q (ZD X ZD)" (with Michael Singer) n 4 o n o o Communications in Algebra, Vol. -
36 Rings of Fractions
36 Rings of fractions Recall. If R is a PID then R is a UFD. In particular • Z is a UFD • if F is a field then F[x] is a UFD. Goal. If R is a UFD then so is R[x]. Idea of proof. 1) Find an embedding R,! F where F is a field. 2) If p(x) 2 R[x] then p(x) 2 F[x] and since F[x] is a UFD thus p(x) has a unique factorization into irreducibles in F[x]. 3) Use the factorization in F[x] and the fact that R is a UFD to obtain a factorization of p(x) in R[x]. 36.1 Definition. If R is a ring then a subset S ⊆ R is a multiplicative subset if 1) 1 2 S 2) if a; b 2 S then ab 2 S 36.2 Example. Multiplicative subsets of Z: 1) S = Z 137 2) S0 = Z − f0g 3) Sp = fn 2 Z j p - ng where p is a prime number. Note: Sp = Z − hpi. 36.3 Proposition. If R is a ring, and I is a prime ideal of R then S = R − I is a multiplicative subset of R. Proof. Exercise. 36.4. Construction of a ring of fractions. Goal. For a ring R and a multiplicative subset S ⊆ R construct a ring S−1R such that every element of S becomes a unit in S−1R. Consider a relation on the set R × S: 0 0 0 0 (a; s) ∼ (a ; s ) if s0(as − a s) = 0 for some s0 2 S Check: ∼ is an equivalence relation.