Differential Resultants
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Dimension Theory and Systems of Parameters
Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite. -
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 -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Effective Noether Irreducibility Forms and Applications*
Appears in Journal of Computer and System Sciences, 50/2 pp. 274{295 (1995). Effective Noether Irreducibility Forms and Applications* Erich Kaltofen Department of Computer Science, Rensselaer Polytechnic Institute Troy, New York 12180-3590; Inter-Net: [email protected] Abstract. Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring, and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial, and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class . The factorization results also extend to the case where the coefficient field is a function field. -
Real Closed Fields
University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1968 Real closed fields Yean-mei Wang Chou The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Chou, Yean-mei Wang, "Real closed fields" (1968). Graduate Student Theses, Dissertations, & Professional Papers. 8192. https://scholarworks.umt.edu/etd/8192 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. EEAL CLOSED FIELDS By Yean-mei Wang Chou B.A., National Taiwan University, l96l B.A., University of Oregon, 19^5 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MONTANA 1968 Approved by: Chairman, Board of Examiners raduate School Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38993 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OwMTtation PVblmhing UMI EP38993 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. -
THE RESULTANT of TWO POLYNOMIALS Case of Two
THE RESULTANT OF TWO POLYNOMIALS PIERRE-LOÏC MÉLIOT Abstract. We introduce the notion of resultant of two polynomials, and we explain its use for the computation of the intersection of two algebraic curves. Case of two polynomials in one variable. Consider an algebraically closed field k (say, k = C), and let P and Q be two polynomials in k[X]: r r−1 P (X) = arX + ar−1X + ··· + a1X + a0; s s−1 Q(X) = bsX + bs−1X + ··· + b1X + b0: We want a simple criterion to decide whether P and Q have a common root α. Note that if this is the case, then P (X) = (X − α) P1(X); Q(X) = (X − α) Q1(X) and P1Q − Q1P = 0. Therefore, there is a linear relation between the polynomials P (X);XP (X);:::;Xs−1P (X);Q(X);XQ(X);:::;Xr−1Q(X): Conversely, such a relation yields a common multiple P1Q = Q1P of P and Q with degree strictly smaller than deg P + deg Q, so P and Q are not coprime and they have a common root. If one writes in the basis 1; X; : : : ; Xr+s−1 the coefficients of the non-independent family of polynomials, then the existence of a linear relation is equivalent to the vanishing of the following determinant of size (r + s) × (r + s): a a ··· a r r−1 0 ar ar−1 ··· a0 .. .. .. a a ··· a r r−1 0 Res(P; Q) = ; bs bs−1 ··· b0 b b ··· b s s−1 0 . .. .. .. bs bs−1 ··· b0 with s lines with coefficients ai and r lines with coefficients bj. -
Are Maximal Ideals? Solution. No Principal
Math 403A Assignment 6. Due March 1, 2013. 1.(8.1) Which principal ideals in Z[x] are maximal ideals? Solution. No principal ideal (f(x)) is maximal. If f(x) is an integer n = 1, then (n, x) is a bigger ideal that is not the whole ring. If f(x) has positive degree, then ± take any prime number p that does not divide the leading coefficient of f(x). (p, f(x)) is a bigger ideal that is not the whole ring, since Z[x]/(p, f(x)) = Z/pZ[x]/(f(x)) is not the zero ring. 2.(8.2) Determine the maximal ideals in the following rings. (a). R R. × Solution. Since (a, b) is a unit in this ring if a and b are nonzero, a maximal ideal will contain no pair (a, 0), (0, b) with a and b nonzero. Hence, the maximal ideals are R 0 and 0 R. ×{ } { } × (b). R[x]/(x2). Solution. Every ideal in R[x] is principal, i.e., generated by one element f(x). The ideals in R[x] that contain (x2) are (x2) and (x), since if x2 R[x]f(x), then x2 = f(x)g(x), which 2 ∈ 2 2 means that f(x) divides x . Up to scalar multiples, the only divisors of x are x and x . (c). R[x]/(x2 3x + 2). − Solution. The nonunit divisors of x2 3x +2=(x 1)(x 2) are, up to scalar multiples, x2 3x + 2 and x 1 and x 2. The− maximal ideals− are− (x 1) and (x 2), since the quotient− of R[x] by− either of those− ideals is the field R. -
APPENDIX 2. BASICS of P-ADIC FIELDS
APPENDIX 2. BASICS OF p-ADIC FIELDS We collect in this appendix some basic facts about p-adic fields that are used in Lecture 9. In the first section we review the main properties of p-adic fields, in the second section we describe the unramified extensions of Qp, while in the third section we construct the field Cp, the smallest complete algebraically closed extension of Qp. In x4 section we discuss convergent power series over p-adic fields, and in the last section we give some examples. The presentation in x2-x4 follows [Kob]. 1. Finite extensions of Qp We assume that the reader has some familiarity with I-adic topologies and comple- tions, for which we refer to [Mat]. Recall that if (R; m) is a DVR with fraction field K, then there is a unique topology on K that is invariant under translations, and such that a basis of open neighborhoods of 0 is given by fmi j i ≥ 1g. This can be described as the topology corresponding to a metric on K, as follows. Associated to R there is a discrete valuation v on K, such that for every nonzero u 2 R, we have v(u) = maxfi j u 62 mig. If 0 < α < 1, then by putting juj = αv(u) for every nonzero u 2 K, and j0j = 0, one gets a non-Archimedean absolute value on K. This means that j · j has the following properties: i) juj ≥ 0, with equality if and only if u = 0. ii) ju + vj ≤ maxfjuj; jvjg for every u; v 2 K1. -
Class Numbers of Number Rings: Existence and Computations
CLASS NUMBERS OF NUMBER RINGS: EXISTENCE AND COMPUTATIONS GABRIEL GASTER Abstract. This paper gives a brief introduction to the theory of algebraic numbers, with an eye toward computing the class numbers of certain number rings. The author adapts the treatments of Madhav Nori (from [1]) and Daniel Marcus (from [2]). None of the following work is original, but the author does include some independent solutions of exercises given by Nori or Marcus. We assume familiarity with the basic theory of fields. 1. A Note on Notation Throughout, unless explicitly stated, R is assumed to be a commutative integral domain. As is customary, we write Frac(R) to denote the field of fractions of a domain. By Z, Q, R, C we denote the integers, rationals, reals, and complex numbers, respectively. For R a ring, R[x], read ‘R adjoin x’, is the polynomial ring in x with coefficients in R. 2. Integral Extensions In his lectures, Madhav Nori said algebraic number theory is misleadingly named. It is not the application of modern algebraic tools to number theory; it is the study of algebraic numbers. We will mostly, therefore, concern ourselves solely with number fields, that is, finite (hence algebraic) extensions of Q. We recall from our experience with linear algebra that much can be gained by slightly loosening the requirements on the structures in question: by studying mod- ules over a ring one comes across entire phenomena that are otherwise completely obscured when one studies vector spaces over a field. Similarly, here we adapt our understanding of algebraic extensions of a field to suitable extensions of rings, called ‘integral.’ Whereas Z is a ‘very nice’ ring—in some sense it is the nicest possible ring that is not a field—the integral extensions of Z, called ‘number rings,’ have surprisingly less structure. -
The Algebraic Closure of a $ P $-Adic Number Field Is a Complete
Mathematica Slovaca José E. Marcos The algebraic closure of a p-adic number field is a complete topological field Mathematica Slovaca, Vol. 56 (2006), No. 3, 317--331 Persistent URL: http://dml.cz/dmlcz/136929 Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 2006 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Mathematíca Slovaca ©20106 .. -. c/. ,~r\r\c\ M -> ->•,-- 001 Mathematical Institute Math. SlOVaCa, 56 (2006), NO. 3, 317-331 Slovák Academy of Sciences THE ALGEBRAIC CLOSURE OF A p-ADIC NUMBER FIELD IS A COMPLETE TOPOLOGICAL FIELD JosÉ E. MARCOS (Communicated by Stanislav Jakubec) ABSTRACT. The algebraic closure of a p-adic field is not a complete field with the p-adic topology. We define another field topology on this algebraic closure so that it is a complete field. This new topology is finer than the p-adic topology and is not provided by any absolute value. Our topological field is a complete, not locally bounded and not first countable field extension of the p-adic number field, which answers a question of Mutylin. 1. Introduction A topological ring (R,T) is a ring R provided with a topology T such that the algebraic operations (x,y) i-> x ± y and (x,y) \-> xy are continuous. -
Ideals and Class Groups of Number Fields
Ideals and class groups of number fields A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Minjiao Yang August, 2018 ○C Copyright All rights reserved Except for previously published materials Thesis written by Minjiao Yang B.S., Kent State University, 2015 M.S., Kent State University, 2018 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematics Science James L. Blank , Dean, College of Arts and Science TABLE OF CONTENTS…………………………………………………………...….…...iii ACKNOWLEDGEMENTS…………………………………………………………....…...iv CHAPTER I. Introduction…………………………………………………………………......1 II. Algebraic Numbers and Integers……………………………………………......3 III. Rings of Integers…………………………………………………………….......9 Some basic properties…………………………………………………………...9 Factorization of algebraic integers and the unit group………………….……....13 Quadratic integers……………………………………………………………….17 IV. Ideals……..………………………………………………………………….......21 A review of ideals of commutative……………………………………………...21 Ideal theory of integer ring 풪푘…………………………………………………..22 V. Ideal class group and class number………………………………………….......28 Finiteness of 퐶푙푘………………………………………………………………....29 The Minkowski bound…………………………………………………………...32 Further remarks…………………………………………………………………..34 BIBLIOGRAPHY…………………………………………………………….………….......36 iii ACKNOWLEDGEMENTS I want to thank my advisor Dr. Gang Yu who has been very supportive, patient and encouraging throughout this tremendous and enchanting experience. Also, I want to thank my thesis committee members Dr. Ulrike Vorhauer and Dr. Stephen Gagola who help me correct mistakes I made in my thesis and provided many helpful advices. iv Chapter 1. Introduction Algebraic number theory is a branch of number theory which leads the way in the world of mathematics. It uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Concepts and results in algebraic number theory are very important in learning mathematics. -
11. Splitting Field, Algebraic Closure 11.1. Definition. Let F(X)
11. splitting field, Algebraic closure 11.1. Definition. Let f(x) F [x]. Say that f(x) splits in F [x] if it can be decomposed into linear factors in F [x]. An∈ extension K/F is called a splitting field of some non-constant polynomial f(x) if f(x) splits in K[x]butf(x) does not split in M[x]foranyanyK ! M F . If an extension K of F is the splitting field of a collection of polynomials in F [x], then⊇ we say that K/F is a normal extension. 11.2. Lemma. Let p(x) be a non-constant polynomial in F [x].Thenthereisafiniteextension M of F such that p(x) splits in M. Proof. Induct on deg(p); the case deg(p) = 1 is trivial. If p is already split over F ,thenF is already a splitting field. Otherwise, choose an irreducible factor p1(x)ofp(x)suchthat deg(p1(x)) 1. By, there is a finite extension M1/F such that p1 has a root α in M1.Sowe can write p≥(x)=(x α)q(x) in M [x]. Since deg(q) < deg(p), by induction, there is a finite − 1 extension M/M1 such that q splits in M.ThenM/F is also finite and p splits in M. ! 11.3. Lemma. Let σ : F L be an embedding of a field F into a field L.Letp(x) F [x] be an irreducible polynomial→ and let K = F [x]/(p(x)) be the extension of of F obtained∈ by adjoining a root of p.Theembeddingσ extends to K if and only if σ(p)(x) has a root in L.