81. Tulane Math Graduate Algebra Qualifying Exam Syllabus Direct sums, *fundamental theorem on finitely generated abelian groups. 2 PJ Hilton and U. Stammbach, A Course in Homological algebra, http://www.math.tulane.edu/graduate/qualifying/algebra.html

82. PinkMonkey.com Algebra Study Guide 4.1 Theorem PinkMonkey.comFree Online algebra Textbook and StudyGuide -The World s largestsource of Free Booknotes/Literature summaries. Hundreds of titles online for http://www.pinkmonkey.com/studyguides/subjects/algebra/chap4/a0404101.asp

Support the Monkey! Tell All your Friends and Teachers Free College Search Click Here Get the "Dirt" on your College! Get a Scholarship! See What's New on the Message Boards today! The Unofficial Guide to College Home ... New Title Request Win a $1000 or more Scholarship to college! Please Take our User Survey WebMasters Click Here CHAPTER 4 : QUADRATIC EQUATIONS If a, b R and ab = then either a = or b = Corollary 1 : then b = Corollary 2 : 0, b then c = Corollary 3 : then either a = or b = c Example Solution : Example Solution : For what values of x is the expression x (3x - 6 ) = ? i.e. x (3x - 6 ) = either x = or 3x -6 = i.e. x = or 3x = 6 x = 2 ( x + 3 ) ( 3x + 1 ) - ( x - 2 ) ( x + 3 ) next page Index 4.1 Theorem 4.2 Definition 4.3 Methods of Solving Quadratic Equations Chapter 5 1340 PinkMonkey users are on the site and studying right now. Search: All Products Books Popular Music Classical Music Video DVD Electronics Software Outdoor Living Cell Phones Keywords:

83. GeoSci 236: The Fundamental Theorem Of Linear Algebra GeoSci 236 The Fundamental theorem of Linear algebra. Gidon Eshel 491 Hinds Dept.of the Geophysical Sciences, 5734 S. Ellis Ave., The Univ. of Chicago, http://geosci.uchicago.edu/~gidon/geosci236/fundam/

GeoSci 236: The Fundamental Theorem of Linear Algebra Gidon Eshel 491 Hinds Dept. of the Geophysical Sciences, 5734 S. Ellis Ave., The Univ. of Chicago, Chicago, IL 60637 geshel@midway.uchicago.edu Figure 1: The forward problem (the fundamental theorem of linear algebra). A 's domain is the upper-left space, while its range is the lower-right one. In the domain, A 's row-space is shown in red , while its nullspace in blue . A generic vector comprising both a row-space and a nullspace components is the vector on which A operates, mapping it onto the adjoint space (lower-right). In the latter space, the shown b comprises components from A 's range (column-space) and left nullspace Figure 1 represents the operation of a matrix on a vector (the upper-left space). That is, it shows schematically what happens when an arbitrary vector from 's domain (the space corresponding dimensionally to 's row dimension N ) is mapped by onto the range space (the space corresponding dimensionally to 's column dimension M ). Hence the schematic shows what happens to from the upper-left space as transforms it to the range, the lower-right space. Put differently, this schematic represents the

84. Contents Of Steinberger's Algebra The Fundamental theorem of algebra 473 11.7. Cyclotomic Extensions 475 11.8. nth Roots 479 11.9. Cyclic Extensions 484 http://math.albany.edu/~mark/algtoc.html

Contents of Algebra Mark Steinberger Chapter 1. A Little Set Theory ... 1 1.1. Properties of Functions ... 1 1.2. Factorizations of Functions ... 3 1.3. Relations ... 6 1.4. Equivalence Relations ... 8 1.5. Generating an Equivalence Relation ... 11 1.6. Cartesian Products ... 12 1.7. Formalities about Functions ... 14 Chapter 2. Groups: Basic Definitions and Examples ... 16 2.1. Groups and Monoids ... 16 2.2. Subgroups ... 20 2.3. The Subgroups of the Integers ... 25 2.4. Finite Cyclic Groups: Modular Arithmetic ... 28 2.5. Homomorphisms and Isomorphisms ... 30 2.6. The Classification Problem ... 36 2.7. The Group of Rotations of the Plane ... 38 2.8. The Dihedral Groups ... 40 2.9. Quaternions ... 42 2.10. Direct Products ... 44 Chapter 3. G-sets and Counting ... 48 3.1. Symmetric Groups: Cayley's Theorem ... 49 3.2. Cosets and Index: Lagrange's Theorem ... 53 3.3. G-sets and Orbits ... 57 3.4. Supports of Permutations ... 66 3.5. Cycle Structure ... 68 3.6. Conjugation and Other Automorphisms ... 73 3.7. Conjugating Subgroups: Normality ... 79

85. Katholieke Universiteit Nijmegen The FTA project (Fundamental theorem of algebra), started in 1999 and to befinished in 2001, has as its main goal to formalize (in Coq) a large body of http://mowgli.cs.unibo.it/html_yes_frames/sites/nijmegen.html

MoWGLI: Mathematics on the Web: Get It by Logic and Interfaces Katholieke Universiteit Nijmegen The Netherlands Subfaculteit Informatica, Faculteit Natuurwetenschappen, Wiskunde en Informatica, Katholieke Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands Visit the institution home page. Site responsible: Prof. Herman Geuvers Site members: Prof. Herman Geuvers Prof. Arjeh Cohen Prof. Henk Barendregt Dr. Freek Wiedijk ... Dan Synek The Sub-faculty of Computer Science at the University of Nijmegen hosts a broad experience in logic, formal methods and theorem proving. The Faculty of Mathematics and Computer Science of Eindhoven University of Technology is strong in computer algebra, theorem proving and applying Web technology to mathematics. Nijmegen and Eindhoven have a long history in cooperation on topics related to this FET proposal, notably type theory, theorem proving and combining various computer mathematics applications, especially using OpenMath. This cooperation was mainly taking place between the research groups of Geuvers and Barendregt in Nijmegen and the research group of Cohen in Eindhoven. The research group of Geuvers and Barendregt is part of the EC sponsored Thematic Network ``TYPES'' (IST-1999-29001) and of its ancestor, the EC Working Group ``Types for Proofs and Programs'', which testifies there interest in theorem proving, especially using type theory based theorem provers. The FTA project (Fundamental Theorem of Algebra), started in 1999 and to be finished in 2001, has as its main goal to formalize (in Coq) a large body of undergraduate mathematics (algebra and analysis), culminating in a proof of the fundamental theorem of algebra. The formalization of the mathematics is now finished and the next step is to make the formalization accessible and usable by others, preferably through the World Wide Web.

86. Wilson Stothers' Cabri Pages - Algebra Proof of theorem 1. We now show that the algebraic polar coincides with the Proof of theorem 2. It follows readily that the algebraic and geometric http://www.maths.gla.ac.uk/~wws/cabripages/algebra.html

Poles, polars and duality -the algebraic version In this section, we give an algebraic treatment of these topics. The proofs may be obtained by clicking on the link below the statement of each theorem. A plane conic has an equation of the form ax +bxy +cy +fx+gy+h=0. In terms of homogeneous coordinates , this becomes ax +bxy +cy +fxz+gyz+hz which can be written as x T M x where x =(x,y,z) , and M is a symmetric 3x3 matrix. For a non-degenerate conic, M must be non-singular and have eigenvalues of different sign. Note that, if a conic contains three (distinct) collinear points, then it must be degenerate. Definition If C: x T M x is a non-degenerate conic and U=[ u is any point, then the algebraic polar of U with respect to C is the line u T M x Note that, as M is non-singular, we cannot have u T M= , so that the line always exists. A line L has an equation a T x . Now, u T M x and a T x give the same line if and only if u ]=[M a Thus L is the polar of a unique point U=[ u Definition If C: x T M x is a non-degenerate conic and L is any line, then the algebraic pole of L with respect to C is the point U=[ u such that L has equation u T M x Remark If L has equation a T x , then, as we have seen, the pole of L is U=[M a Theorem 1 If C: x T M x is a non-degenerate conic and U is any point on C then the algebraic polar of U with respect to C is the tangent to C at U Proof of Theorem 1 We now show that the algebraic polar coincides with the geometrical polar.

87. Matrix Reference Manual This manual contains reference information about linear algebra and the If some matrices are of a particular form, the theorem is prefaced thus A http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html

The Matrix Reference Manual Mike Brookes , Imperial College, London, UK Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled " GNU Free Documentation License To cite this manual use: Brookes, M., "The Matrix Reference Manual", [online] http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html, 2005 Introduction This manual contains reference information about linear algebra and the properties of real and complex matrices. The manual is divided into the following sections: Main Index : Alphabetical index of all entries. Properties Eigenvalues : Theorems and matrix properties relating to eigenvalues and eigenvectors. Special Relations Decompositions : Decomposing matrices as sums or products of simpler forms. Identities : Useful equations relating matrices. Equations : Solutions of matrix equations Calculus : Differentiating expressions involving matrices whose elements are functions of an independent variable.

88. [hep-th/0305087] Basic Theorem, Gauge Algebra, $\theta$-superfield QED In The La Basic theorem, Gauge algebra, $\theta$superfield QED in the Lagrangian Formulationof General Superfield Theory of Fields. Authors AA Reshetnyak (Seversk http://arxiv.org/abs/hep-th/0305087

High Energy Physics - Theory, abstract hep-th/0305087 From: Alexander A. Reshetnyak [ view email ] Date ( ): Sun, 11 May 2003 08:55:13 GMT (25kb) Date (revised v2): Wed, 21 May 2003 11:50:38 GMT (25kb) Authors: A.A. Reshetnyak (Seversk State Technological Institute) Comments: 21 pages, Latex 2e, no figures, minor corrections before formula (6.28) and in reference [9] The basic theorem of the Lagrangian formulation for general superfield theory of fields (GSTF) is proved. The gauge transformations of general type (GTGT) and gauge algebra of generators of GTGT (GGTGT) as the consequences of the above theorem are studied. It is established the gauge algebra of GGTGT contains the one of generators of gauge transformations of special type (GGTST) as one's subalgebra. In the framework of Lagrangian formulation for GSTF the nontrivial superfield model generalizing the model of Quantum Electrodynamics and belonging to the class of gauge theory of general type (GThGT) with Abelian gauge algebra of GGTGT is constructed. Full-text: PostScript PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited

89. Robbins Algebras Are Boolean A web text by William McCune describing the solution of this problem by a theoremproving program, with input files and the proofs. http://www-unix.mcs.anl.gov/~mccune/papers/robbins/

Robbins Algebras Are Boolean William McCune Automated Deduction Group Mathematics and Computer Science Division Argonne National Laboratory Posted on the Web October 15, 1996. Last updated September 24, 2003. These Web pages contain some information on the solution of the Robbins problem. A paper on this topic appears in the Journal of Automated Reasoning [W. McCune, "Solution of the Robbins Problem", JAR 19(3), 263276 (1997)]. Here is a preprint . The JAR paper has simpler proofs than the ones below on this page. Here are the input files and proofs corresponding to the JAR paper A draft of a press release , intended for a wider audience, is also available. Introduction The Robbins problem-are all Robbins algebras Boolean?-has been solved: Every Robbins algebra is Boolean. This theorem was proved automatically by EQP , a theorem proving program developed at Argonne National Laboratory. Historical Background In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra: x + y = y + x. [commutativity] (x + y) + z = x + (y + z). [associativity] n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation] Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one [5]:

90. Renate Schmidt: Home Page University of Manchester Modal logic, resolution theorem proving, resolution decision problems, relation algebras, Peirce algebras and knowledge representation. http://www.cs.man.ac.uk/~schmidt/

Dr.-Ing. Renate A. Schmidt Renate A. Schmidt School of Computer Science University of Manchester Oxford Rd, Manchester M13 9PL, UK Tel: +44 (0)161 275 6163 Fax: +44 (0)161 275 6204 Mobile: +44 (0)776 193 5696 Email: Renate.Schmidt@manchester.ac.uk Kilburn Building, Room: 2.42. Research group: Formal Methods (FM) Special Issue on "Implementation of Logics" Upcoming event: Relations and Kleene Algebra in Computer Science Consulation hours Publications Publications ordered by publication date Publications ordered by theme Research Administrative activities Programme director of MSc in Advanced Computer Science with ICT Management Programme director of MSc in Computer Science Professional activities Teaching Currently I am involved with teaching the following course units. CS612 - Automated Reasoning CS616 - Knowledge Representation and Reasoning CS1112 - Reasoning about Programs CS510 - Introduction to Programming Using C Lists of student projects can be found here: Third year projects 2005/2006 MSc projects 2004/2005 PhD projects Students interested in doing a PhD in areas related to automated reasoning, logic, knowledge representation, logic-based multi-agent systems and any topics close to my

91. ABSTRACT ALGEBRA ON LINE Contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. Intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. http://www.math.niu.edu/~beachy/aaol/

WELCOME TO ABSTRACT ALGEBRA ON LINE This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the books Abstract Algebra , by John A. Beachy and William D. Blair, and Abstract Algebra II , by John A. Beachy. The site is organized by chapter. The page containing the Table of Contents also contains an index of definitions and theorems, which can be searched for detailed references on subject area pages. Topics from the first volume are marked by the symbol and those from the second volume by the symbol . To make use of this site as a reference, please continue on to the Table of Contents. TABLE OF CONTENTS (No frames) TABLE OF CONTENTS (Frames version) Interested students may also wish to refer to a closely related site that includes solved problems: the OnLine Study Guide for Abstract Algebra REFERENCES Abstract Algebra Second Edition , by John A. Beachy and William D. Blair

92. Digital Logic There is a group of useful theorems of Boolean algebra which help in developing the These theorems can be used in the algebraic simplification of logic http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/diglog.html

Digital Logic For two binary variables (taking values and 1) there are 16 possible functions . The functions involve only three operations which make up Boolean algebra: AND, OR, and COMPLEMENT. They are symbolically represented as follows: These operations are like ordinary algebraic operations in that they are commutative associative , and distributive . There is a group of useful theorems of Boolean algebra which help in developing the logic for a given operation. Digital Logic Theorems Digital Logic Functions Index Electronics concepts ... Electricity and magnetism R Nave Go Back Boolean Algebra Theorems The applications of digital logic involve functions of the AND, OR, and NOT operations. These operations are subject to the following identities: These theorems can be used in the algebraic simplification of logic circuits which come from a straightforward application of a truth table DeMorgan's Theorem Basic Gates Index ... Electricity and magnetism R Nave Go Back Binary Functions of Two Variables Digital logic involves combinations of the three types of operations for two variables: AND, OR, and NOT. There are sixteen possible functions: This is an active graphic. Click on any of the functions for further details.

93. Boolean Algebra In this class we will use the axioms and theorems of Boolean algebra to simplifyBoolean expressions. Using Boolean algebra to simplify Boolean expressions http://134.193.15.25/vu/course/cs281/lectures/boolean-algebra/boolean-algebra.ht

Boolean Algebra Introduction You may have been intimidated by the mathematical word problems on the SAT or ACT. On first reading they seem almost impossible to solve. "Larry and Fred share a bookcase. Larry has twice as many books as Fred. There are 75 books on the bookcase. How many books does Fred have?" Then you realize how the problem can be redefined as an algebra problem. As an algebra problem the solution is much easier. 2*F + F = 75 F = 25 Fred has 25 books. For many of the same reasons digital systems are based on an algebranot the regular algebra you and I are familiar with but rather Boolean algebra. Boolean algebra is the theoretical foundation for digital systems. Boolean algebra formalizes the rules of logic. On the surface computers are great number crunchers, but inside computations are performed by binary digital circuits following the rules of logic. We use Boolean algebra in this class to simplify Boolean expressions which represent circuits. In this lecture we will study algebraic techniques for simplifying expressions. In the next lecture we will look at mechanical waysalgorithms you can use with pencil and paper to simplify moderately complex Boolean functions and algorithms that machines can follow to simplify arbitrarily complex Boolean functions. Axioms In 1854 George Boole Introduced the following formalism that eventually became Boolean Algebra.

94. DeMorgan's Theorems - Chapter 7: BOOLEAN ALGEBRA - Volume IV - Digital DeMorgan s theorems state the same equivalence in backward form that invertingthe properties, rules, and theorems (DeMorgan s) of Boolean algebra http://www.allaboutcircuits.com/vol_4/chpt_7/8.html

Volume I - DC Volume II - AC Volume III - Semiconductors Volume IV - Digital ... Back to Chapter Index Search All Volumes Volume I - DC Volume II - AC Volume III - Semiconductors Volume IV - Digital Volume V - Reference Volume VI - Experiments Check out our new Electronics Forums Ask questions and help answer others. Check it out! DeMorgan's Theorems All About Circuits Volume IV - Digital Chapter 7: BOOLEAN ALGEBRA DeMorgan's Theorems DeMorgan's Theorems A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra. By group complementation, I'm referring to the complement of a group of terms, represented by a long bar over more than one variable. You should recall from the chapter on logic gates that inverting all inputs to a gate reverses that gate's essential function from AND to OR, or visa-versa, and also inverts the output. So, an OR gate with all inputs inverted (a Negative-OR gate) behaves the same as a NAND gate, and an AND gate with all inputs inverted (a Negative-AND gate) behaves the same as a NOR gate. DeMorgan's theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs: A long bar extending over the term AB acts as a grouping symbol, and as such is entirely different from the product of A and B independently inverted. In other words, (AB)' is not equal to A'B'. Because the "prime" symbol (') cannot be stretched over two variables like a bar can, we are forced to use parentheses to make it apply to the whole term AB in the previous sentence. A bar, however, acts as its own grouping symbol when stretched over more than one variable. This has profound impact on how Boolean expressions are evaluated and reduced, as we shall see.

95. Mathematics Archives - Topics In Mathematics - Abstract Algebra KEYWORDS Definitions, Theorems; algebra 1 ADD. KEYWORDS Mathematica Notebooks,groups, linear groups, cosets, normal subgroups, quotient groups, http://archives.math.utk.edu/topics/abstractAlgebra.html

Topics in Mathematics Abstract Algebra Abstract Algebra ADD. KEYWORDS: Book SOURCE: John A. Beachy from Northern Illinois University Abstract Algebra ADD. KEYWORDS: Course materials, Lecture Notes SOURCE: Frank W. Anderson, University of Oregon TECHNOLOGY: TeX Abstract Algebra On Line ADD. KEYWORDS: Definitions, Theorems Algebra 1 ADD. KEYWORDS: Mathematica Notebooks, groups, linear groups, cosets, normal subgroups, quotient groups, group actions and Sylow subgroups SOURCE: Scarborough College Mathematics Computer Laboratory TECHNOLOGY: Mathematica Algebra: Abstract and Concrete by Frederick M. Goodman ADD. KEYWORDS: Published Text, Supplementary Materials, VRML, Symmetry, Polyhedra, Duality of polyhedra, Symmetric dynamical systems, Paper and stick models Algebra Universalis ADD. KEYWORDS: Published Journal Algebraic Combinatorics Via Finite Group Actions ADD. KEYWORDS: Online book SOURCE: A. Betten, H. Fripertinger, and A. Kerber The Algebraic Geometry Notebooks for Non-experts ADD. KEYWORDS: Expository article SOURCE: Aksel Sogstad from the University of Oslo Algebraic Geometry Seminar ADD. KEYWORDS:

96. Pythagoras' Theorem - Dissections On the right we have the two smaller squares of Pythagoras theorem and the same The algebraic argument above leads to a curious dissection of the large http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagorasdissecti

Proofs of Pythagoras' Theorem using translations Conceptually, one of the most attractive ways to' prove Pythagoras' Theorem is to find a partition the three squares into smaller regions, with the property that the partition of the large square is assembled from regions in the partitions of the two other squares. Such proofs are among the oldest known. Some of them turn out to be closely related to the straightforward `algebraic' proofs. There is in fact a general and elementary result about areas in the plane (apparently proven first around 1900) which asserts that if two polygonal figures in the plane have the same area, then one can find piecewise congruent decomposition of the two figures. That is to say, on grounds of general principle one knows that if Pythagoras' Theorem is true, one can find the sort of partitions we are looking for. Reference Heinz Hopf, Springer Lecture Notes in Mathematics #1000. David Hilbert, Foundations of Geometry , Open House, 1994. Proof by algebra This has apparently been found independently many times. On each side we have a square of side a + b . On the left inscribed in it is a square of side c , and four copies of the original right triangle. The area of the square on the left is therefore

97. JHU Graduate Program - Courses Affine varieties and commutative algebra. Hilbert s theorems about polynomialsin several variables with their connections to geometry. http://mathnt.mat.jhu.edu/mathnew/gradprogcourses.html

Welcome Awards Basic Program Choosing an Advisor ... TA Responsibilities Graduate Courses Algebra Strategies for Computer-Assisted Mathematics Instruction Real Variables Complex Variables Several Complex Variables Algebraic Topology Number Theory Lie Groups and Lie Algebras Partial Differential Equations Microlocal Analysis Algebraic Geometry Riemannian Geometry Quantum Cohomology Homotopy Theory Topics in Automorphic Functions Topics in Analysis Topics in Algebraic Topology Topics in Several Complex Variables Topics in Algebraic Number Theory Topics in Hodge Theory Topics in Algebraic Geometry Topics in Partial Differential Equations Topics in Group Representations JAMI Seminar Thesis Research Independent Study, Graduate Algebra An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for Number Theory, Algebraic Geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras. Prerequisites Strategies for Computer-Assisted Mathematics Instruction This course is designed to introduce teaching assistants to the Maple program and to explore strategies using Maple in the teaching of undergraduate mathematics. It may be required as a part of their normal duties. Others may enroll only with permission of instructor.

98. Read This: Logic As Algebra The MAA Online book review column reviews of Logic as algebra by Halmos and Givant . of the subject and wellknown theorems in algebra becomes clearer. http://www.maa.org/reviews/logalg.html

Search MAA Online MAA Home Read This! The MAA Online book review column Logic as Algebra by Paul Halmos and Steven Givant Reviewed by Mark Johnson The intent of Logic as Algebra is expressed clearly in its preface: ...to show that logic can (and perhaps should) be viewed from an algebraic perspective. When so viewed, many of its principal notions are seen to be old friends, familiar algebraic notions that were "disguised" in logical clothing. Moreover, the connection between the principal theorems of the subject and well-known theorems in algebra becomes clearer. Even the proofs often gain in simplicity. To this end, authors Paul Halmos and Steven Givant have written a brief, engaging text aimed at both amateurs and professionals, which requires only a course in modern algebra as background. Since one of this book's potential uses is as a course text, an outline of its contents may be helpful: What is Logic? An introductory parable and mini-logic prepare the reader to meet propositional logic. Propositional Calculus A fairly traditional development of propositional logic.

99. Algebra I+II Syllabus algebra I Syllabus. Finitely generated modules over PID s, the structuretheorem. References algebra (3rd Edition), Lang, 1993, AddisonWesley, http://www.math.sunysb.edu/graduate/algebra.i.ii/

MAT 534 and 535 Algebra I + II Prerequisites A year of undergraduate algebra, such as MAT 313 and MAT 318 . Thus basic notions concerning set theory, cardinals, ordinals, prime numbers, Euclidean algorithm, congruences, polynomials, complex numbers, abelian and cyclic groups, permutation groups, rings and fields, vector spaces are assumed or briefly reviewed. A good reference is Algebra by Michael Artin, Prentice Hall, 1991. Algebra I (Fall) Groups (5 weeks) Direct products, Normal subgroups, Quotient groups, and the isomorphism theorems. Groups acting on sets; orbits and stabilizers. Applications: class formula, centralizers and normalizers, centers of finite p-groups. Conjugacy classes of S n Sylow's Theorems, Solvable groups, Simple groups, simplicity of A n . Examples: Finite groups of small order ( Structure of finitely generated abelian groups. Free groups. Applications. References: Algebra (3rd Edition), Lang, 1993, Addison-Wesley, chapter I. Abstract Algebra (2nd edition), Dummit and Foote, 1999, Part I. Introduction to the Theory of Groups, Rotman, Springer Verlag. Basic linear algebra (3 weeks) Vector spaces, Linear dependence/independence, Bases, Matrices and linear maps. Dual vector space, quotient vector spaces, isomorphism theorems.

100. Applied Abstract Algebra Applied Abstract algebra. Pascal s triangle revisited The FundamentalTheorem of Arithmetic Perfect numbers and Mersenne primes Primality testing http://web.usna.navy.mil/~wdj/book/

Next: Contents Contents Index Applied Abstract Algebra D. Joyner, R. Kreminski, J. Turisco Date: ROUGH DRAFT Please see book page for more details on this book (errata, some maple-related exercises, etc). Contents Some elementary number theory Introduction Divisibility ... Binary and -ary notation Application: Divisibility criteria Application: Winning the game of Nim The Euclidean algorithm Ideals ... An application to Euler's -function Integral powers mod Fermat's Little Theorem, revisited Euler's theorem Application: Decimal expansions of rational numbers ... Arithmetic properties of : a summary Special project: continued fractions (optional) Number theory exercises using GAP Number theory exercises using MAGMA Polynomials, rings and fields ... Polynomials is a ring Roots The division algorithm The Euclidean algorithm Extended Euclidean Algorithm ... Factoring over Factoring over a finite field Modular arithmetic with polynomials Arithmetic properties of Constructing finite extensions of fields ... Factoring over Special project: Factoring over Special Project: Factoring over or Primitive polynomials Factoring strategies Irreducibility criteria Factoring over Polynomials and rings using GAP Finite fields Some vector spaces Rings ... The Hamming metric -codes and error correction The binary Hamming code Equivalent codes Bounds on the parameters of a code The dual code Computing the check matrix and the encoding matrix ... Question: What is ``the best'' code?

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