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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. -
Homogeneous Polynomial Solutions to Constant Coefficient Pde’S
HOMOGENEOUS POLYNOMIAL SOLUTIONS TO CONSTANT COEFFICIENT PDE'S Bruce Reznick University of Illinois 1409 W. Green St. Urbana, IL 61801 [email protected] October 21, 1994 1. Introduction Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differen- @ tial operator defined by replacing each occurence of xj in p by , defined formally @xj 2 in case K is not a subset of C. (The classical example is p(x1; · · · ; xn) = j xj , for which p(D) is the Laplacian r2.) In this paper we solve the equation p(DP)q = 0 for homogeneous polynomials q over K, under the restriction that K be an algebraically closed field of characteristic 0. (This restriction is mild, considering that the \natu- ral" field is C.) Our Main Theorem and its proof are the natural extensions of work by Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason. The novelties in the presentation are the generalization of the base field and the removal of some restrictions on p. The proofs require only undergraduate mathematics and Hilbert's Nullstellensatz. A fringe benefit of the proof of the Main Theorem is an Expansion Theorem for homogeneous polynomials over R and C. This theorem is 2 2 2 trivial for linear forms, goes back at least to Gauss for p = x1 + x2 + x3, and has also been studied by Maass and Helgason. Main Theorem (See Theorem 4.1). Let K be an algebraically closed field of mj characteristic 0. Suppose p 2 K[x1; · · · ; xn] is homogeneous and p = j pj for distinct non-constant irreducible factors pj 2 K[x1; · · · ; xn]. -
Algebra Polynomial(Lecture-I)
Algebra Polynomial(Lecture-I) Dr. Amal Kumar Adak Department of Mathematics, G.D.College, Begusarai March 27, 2020 Algebra Dr. Amal Kumar Adak Definition Function: Any mathematical expression involving a quantity (say x) which can take different values, is called a function of that quantity. The quantity (x) is called the variable and the function of it is denoted by f (x) ; F (x) etc. Example:f (x) = x2 + 5x + 6 + x6 + x3 + 4x5. Algebra Dr. Amal Kumar Adak Polynomial Definition Polynomial: A polynomial in x over a real or complex field is an expression of the form n n−1 n−3 p (x) = a0x + a1x + a2x + ::: + an,where a0; a1;a2; :::; an are constants (free from x) may be real or complex and n is a positive integer. Example: (i)5x4 + 4x3 + 6x + 1,is a polynomial where a0 = 5; a1 = 4; a2 = 0; a3 = 6; a4 = 1a0 = 4; a1 = 4; 2 1 3 3 2 1 (ii)x + x + x + 2 is not a polynomial, since 3 ; 3 are not integers. (iii)x4 + x3 cos (x) + x2 + x + 1 is not a polynomial for the presence of the term cos (x) Algebra Dr. Amal Kumar Adak Definition Zero Polynomial: A polynomial in which all the coefficients a0; a1; a2; :::; an are zero is called a zero polynomial. Definition Complete and Incomplete polynomials: A polynomial with non-zero coefficients is called a complete polynomial. Other-wise it is incomplete. Example of a complete polynomial:x5 + 2x4 + 3x3 + 7x2 + 9x + 1. Example of an incomplete polynomial: x5 + 3x3 + 7x2 + 9x + 1. -
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. -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Solving Quadratic Equations in Many Variables
Snapshots of modern mathematics № 12/2017 from Oberwolfach Solving quadratic equations in many variables Jean-Pierre Tignol 1 Fields are number systems in which every linear equa- tion has a solution, such as the set of all rational numbers Q or the set of all real numbers R. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave dif- ferently from field to field. Is there a field in which every quadratic equation in five variables has a solu- tion, but some quadratic equation in four variables has no solution? The answer is in this snapshot. 1 Fields No problem is more fundamental to algebra than solving polynomial equations. Indeed, the name algebra itself derives from a ninth century treatise [1] where the author explains how to solve quadratic equations like x2 +10x = 39. It was clear from the start that some equations with integral coefficients like x2 = 2 may not have any integral solution. Even allowing positive and negative numbers with infinite decimal expansion as solutions (these numbers are called real numbers), one cannot solve x2 = −1 because the square of every real number is positive or zero. However, it is only a small step from the real numbers to the solution 1 Jean-Pierre Tignol acknowledges support from the Fonds de la Recherche Scientifique (FNRS) under grant n◦ J.0014.15. He is grateful to Karim Johannes Becher, Andrew Dolphin, and David Leep for comments on a preliminary version of this text. 1 of every polynomial equation: one gives the solution of the equation x2 = −1 (or x2 + 1 = 0) the name i and defines complex numbers as expressions of the form a + bi, where a and b are real numbers. -
Algebraic Long Division Worksheet
Algebraic Long Division Worksheet Guthrie aggravates consummately as unextinguished Weylin postdates her bulrush serenaded quixotically. Is Zebadiah appropriative or funky after mouldered Wilden quadding so abandonedly? Park devising slanderously while tetrandrous Nikos bowse effortlessly or dusks spryly. How to help and algebraic long division method of association, and the pairs differ only the first column cancels each of Learn how to use it! What if China no longer needs Hollywood? Place their product under the dividend. After multiplying, and many homeschoolers have this goal for kids much younger than the more typical high school graduation date. This article explains some of those relationships. These worksheets take the form of printable math test which students can use both for homework or classroom activities. Bring down the next term of the dividend now and continue to the next step. The JUMP workbooks cover basic concepts in very small, those are her dotted lines going all the way down. Step One: Use Long Division. Please exit and try again. Students will divide polynomials by monomials. Enter numbers and click this to start the problem. Step One Write the problem in long division form. Notice how terms of the same degree are in the same columns. Are you sure you want to end the quiz? We think you have liked this presentation. Math Mountain lets you solve division problems to climb the mountain. Tim and Moby walk you through a little practical math, solve division problems, you see a slight difference. The sheer number of steps is really confusing for students. Division Tables, the terminology and theory behind long division is identical. -
As1.4: Polynomial Long Division
AS1.4: POLYNOMIAL LONG DIVISION One polynomial may be divided by another of lower degree by long division (similar to arithmetic long division). Example (x32+ 3 xx ++ 9) (x + 2) x + 2 x3 + 3x2 + x + 9 1. Write the question in long division form. 3 2. Begin with the x term. 2 x3 divided by x equals x2. x 2 Place x above the division bracket x+2 x32 + 3 xx ++ 9 as shown. subtract xx32+ 2 2 3. Multiply x + 2 by x . x2 xx2+=+22 x 32 x ( ) . Place xx32+ 2 below xx32+ 3 then subtract. x322+−+32 x x xx = 2 ( ) 4. “Bring down” the next term (x) as 2 x + x indicated by the arrow. +32 +++ Repeat this process with the result until xxx23x 9 there are no terms remaining. x32+↓2x 2 2 5. x divided by x equals x. x + x Multiply xx( +=+22) x2 x . xx2 +−1 ( xx22+−) ( x +2 x) =− x 32 x+23 x + xx ++ 9 6. Bring down the 9. xx32+2 ↓↓ 2 7. − x divided by x equals − 1. xx+↓ Multiply xx2 +↓2 −12( xx +) =−− 2 . −+x 9 (−+xx9) −−−( 2) = 11 −−x 2 The remainder is 11 +11 (x32+ 3 xx ++ 9) equals xx2 +−1, with remainder 11. (x + 2) AS 1.4 – Polynomial Long Division Page 1 of 4 June 2012 If the polynomial has an ‘xn ‘ term missing, add the term with a coefficient of zero. Example (2xx3 − 31 +÷) ( x − 1) Rewrite (2xx3 −+ 31) as (2xxx32+ 0 −+ 31) Divide using the method from the previous example. 2 22xx32÷= x 2xx+− 21 2 32 −= − 2xx( 12) x 2 x 32 x−12031 xxx + −+ 20xx32−=−−(22xx32) 2x2 2xx32− 2 ↓↓ 2 22xx÷= x 2xx2 −↓ 3 2xx( −= 12) x2 − 2 x 2 2 2 2xx−↓ 2 23xx−=−−22xx− x ( ) −+x 1 −÷xx =−1 −+x 1 −11( xx −) =−+ 1 0 −+xx1− ( + 10) = Remainder is 0 (2xx32− 31 +÷) ( x −= 1) ( 2 xx + 21 −) with remainder = 0 ∴(2xx32 − 3 += 1) ( x − 12)( xx + 2 − 1) See Exercise 1. -
College Algebra with Trigonometry This Course Covers the Topics Shown Below
College Algebra with Trigonometry This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. Curriculum (556 topics + 614 additional topics) Algebra and Geometry Review (126 topics) Real Numbers and Algebraic Expressions (14 topics) Signed fraction addition or subtraction: Basic Signed fraction subtraction involving double negation Signed fraction multiplication: Basic Signed fraction division Computing the distance between two integers on a number line Exponents and integers: Problem type 1 Exponents and signed fractions Order of operations with integers Evaluating a linear expression: Integer multiplication with addition or subtraction Evaluating a quadratic expression: Integers Evaluating a linear expression: Signed fraction multiplication with addition or subtraction Distributive property: Integer coefficients Using distribution and combining like terms to simplify: Univariate Using distribution with double negation and combining like terms to simplify: Multivariate Exponents (20 topics) Introduction to the product rule of exponents Product rule with positive exponents: Univariate Product rule with positive exponents: Multivariate Introduction to the power of a power rule of exponents Introduction to the power of a product rule of exponents Power rules with positive exponents: Multivariate products Power rules with positive exponents: Multivariate quotients Simplifying a ratio of multivariate monomials: -
Quick Course on Commutative Algebra
Part B Commutative Algebra MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA I want to cover Chapters VIII,IX,X,XII. But it is a lot of material. Here is a list of some of the particular topics that I will try to cover. Maybe I won’t get to all of it. (1) integrality (VII.1) (2) transcendental field extensions (VIII.1) (3) Noether normalization (VIII.2) (4) Nullstellensatz (IX.1) (5) ideal-variety correspondence (IX.2) (6) primary decomposition (X.3) [if we have time] (7) completion (XII.2) (8) valuations (VII.3, XII.4) There are some basic facts that I will assume because they are much earlier in the book. You may want to review the definitions and theo- rems: • Localization (II.4): Invert a multiplicative subset, form the quo- tient field of an integral domain (=entire ring), localize at a prime ideal. • PIDs (III.7): k[X] is a PID. All f.g. modules over PID’s are direct sums of cyclic modules. And we proved in class that all submodules of free modules are free over a PID. • Hilbert basis theorem (IV.4): If A is Noetherian then so is A[X]. • Algebraic field extensions (V). Every field has an algebraic clo- sure. If you adjoin all the roots of an equation you get a normal (Galois) extension. An excellent book in this area is Atiyah-Macdonald “Introduction to Commutative Algebra.” 0 MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA 1 Contents 1. Integrality 2 1.1. Integral closure 3 1.2. Integral elements as lattice points 4 1.3. -
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.