To learn more about a topic listed below, click the topic name to go to the corresponding classroom page. | Abstract algebra is the set of advanced topics in algebra that deal with abstract algebraic structures rather than the usual number systems. | | The zero set of a collection of polynomials. An algebraic variety is one of the the fundamental objects in algebraic geometry. | | A Boolean algebra is an algebra where the multiplication and addition also satisfy the properties of the AND and OR operations from logic. | | A category is an abstract mathematical object that generalizes the ideas of maps and commutative diagrams. | | An isomorphism is a map between mathematical objects such as groups, rings, or fields that is one-to-one, onto, and preserves the properties of the object. | | A Lie algebra is a nonassociative algebra corresponding to a Lie group. | | A Lie group is a differentiable manifold that has the structure of a group and that satisfies the additional condition that the group operations of multiplication and inversion are continuous. | Group Theory : | An Abelian group is a group for which the binary operation is commutative. | : | A cyclic graph is an (always Abelian) abstract group generated by a single element. | : | The dihedral group of order is the symmetry group for a regular polygon with sides. | : | A finite group is a group with a finite number of elements. | : | A mathematical group is a set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. | : | A group action is the association of each of a mathematical group's elements with a permutation of the elements of a set. | : | A group representation is a mathematical group action on a vector space. | : | Group theory is the mathematical study of abstract groups, namely sets of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. | : | A normal subgroup is a subgroup that is fixed under conjugation by any element. | : | A simple group is a mathematical group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. | : | A subgroup is a subset of a mathematical group that is also a group. | : | A symmetric group is a group of all permutations of a given set. | : | A symmetry group is a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions. | Rings and Fields : | (1) Algebra is a subject taught in grade school and high school, sometimes referred to as "arithmetic", that includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. (2) In higher mathematics, the term algebra generally refers to abstract algebra, which involves advanced topics that deal with abstract algebraic structures rather than the usual number systems. (3) In topology, an algebra is a vector space that also possesses a vector multiplication. | : | An algebraic number is a number that is the root of some polynomial with integer coefficients. Algebraic numbers can be real or complex and need not be rational. | : | A field is a ring in which every nonzero element has a multiplicative inverse. The real numbers and the complex numbers are both fields. | : | A finite field is a field with a finite number of elements. In such a field, the number of elements is always a power of a prime. | : | A Gaussian integer is a complex number + , where and are integers and is the imaginary unit. | : | In mathematics, and ideal is a subset of a ring that is closed under addition and multiplication by any element of the ring. | : | A module is a generalization of a vector space in which the scalars form a ring rather than a field. | : | A quaternion is a member of a four-dimensional noncommutative division algebra (i.e., a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative) over the real numbers. | : | In mathematics, a ring is an Abelian group together with a rule for multiplying its elements. | Advertisement Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction- Published: 01 January 2016
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Cite this article436 Accesses 38 Citations 1 Altmetric Explore all metrics This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics—and their progression across elementary, middle, and secondary mathematics— where teaching may be transformed by teachers’ knowledge of abstract algebra are developed. In each of the four content areas (arithmetic properties, inverses, structure of sets, and solving equations), descriptions and examples of the transformational influence on teaching these topics are used to depict and support ways that study of more advanced mathematics can influence teachers’ practice. Implications for the mathematical preparation and professional development of teachers are considered. Cet article se penche sur l’influence potentielle de certains aspects de l’algèbre abstraite sur l’enseignement de l’algèbre scolaire (et l’algèbre élémentaire). En utilisant les normes nationales d’analyse, on développe quatre domaines primaires communs dans les mathématiques scolaires, ainsi que leur évolution au travers des classes de mathématiques élémentaires, intermédiaires et secondaires, lorsque l’enseignement peut être modifié par les connaissances de l’enseignant en algèbre abstraite. Dans chacun des quatre domaines (propriétés arithmétiques, inverses, structure des ensembles et résolution d’équations), des descriptions et des exemples de l’influence transformationnelle sur l’enseignement de ces sujets sont utilisés pour décrire et soutenir l’idée que l’étude de mathématiques plus avancées peut influencer la pratique de l’enseignant. Les conséquences pour la préparation mathématique et le perfectionnement professionnel des enseignants sont examinées. This is a preview of subscription content, log in via an institution to check access. Access this articleSubscribe and save. - Get 10 units per month
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Price includes VAT (Russian Federation) Instant access to the full article PDF. Rent this article via DeepDyve Institutional subscriptions Similar content being viewed by othersFrom Equations to Structures: Modes of Relevance of Abstract Algebra to School Mathematics as Viewed by Teacher Educators and TeachersExploring advanced mathematics courses and content for secondary mathematics teachers, what kind of opportunities do abstract algebra courses provide for strengthening future teachers’ mathematical knowledge for teaching. Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education , 6 , 1–32. Article Google Scholar Ball, D. L. (2009, April 23). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures . Paper presented at the National Council of Teachers of Mathematics (NCTM) Annual Meeting, Washington, DC. Retrieved from http://www-personal.umich.edu /∼dball/presentations/index.html Google Scholar Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education , 59 (5), 389–407. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebra reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669–705). Charlotte, NC: Information Age. Common Core State Standards in Mathematics (CCSS-M). (2010). Common Core State Standards in Mathematics . Retrieved from http://www.corestandards.org /the-standards/mathematics Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (MET I) . Retrieved from http://cbmsweb.org /MET_Document/ Book Google Scholar Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II (MET II) . Retrieved from http://www.cbmsweb.org /MET2/MET2Draft.pdf Cuoco, A., & Rotman, J. (2013). Learning modern algebra: From modern attempts to prove Fermat’s last theorem . Washington, DC: Mathematical Association of America. Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives , 8 (1). Retrieved from http://epaa.asu.edu /ojs/article/view/392 Harry, B., Sturges, K., & Klingner, J. K. (2005). Mapping the process: An exemplar of process and challenge in grounded theory analysis. Educational Researcher , 4 (2), 3–13. Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111–155). Charlotte, NC: Information Age. Jakobsen, A., Thames, M. H., & Ribeiro, C. M. (2013, February 5-10). Delineating issues related to horizon content knowledge for mathematics teaching . Paper presented at the Eighth Congress of European Research in Mathematics Education (CERME-8), Manavgat-Side, Antalya-Turkey. Retrieved from http://cerme8.metu.edu.tr /wgpapers/WG17/WG17_Jakobsen_Thames_Ribeiro.pdf Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Lawrence Erlbaum. Klein, F. (1932). Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (E. R. Hedrick & C. A. Noble, Trans.). Mineola, NY: Macmillan. McRory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D., & Senk, S. L. (2012). Knowledge of algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education , 43 (5), 584–615. Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review , 13 (2), 125–145. Morris, A. (1999). Developing concepts of mathematical structure: Pre-arithmetic reasoning versus extended arithmetic reasoning. Focus on Learning Problems in Mathematics , 21 (1), 44–67. Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Educational Research Journal , 40 (4), 905–928. Piaget, J. (1952). The origins of intelligence in children . New York, NY: International University Press. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education , 8 (3), 255–281. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher , 15 (2), 4–14. Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education , 11 (6), 499–511. Simon, M. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning , 8 (4), 359–371. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques . Newbury Park, CA: Sage Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K-12 mathematics teachers. PRIMUS , 24 (3), 191–214. Wasserman, N., & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics , 33 (3), 20–22. Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning , 12 (4), 263–281. Zazkis, R., & Mamolo, A. (2011). Reconceptualizing knowledge at the mathematical horizon. For the Learning of Mathematics , 1 (2), 8–13. Download references Author informationAuthors and affiliations. Teachers College, Columbia University, New York, 525 West 120th Street, Box 210-M, New York, 10027, USA Nicholas H. Wasserman You can also search for this author in PubMed Google Scholar Corresponding authorCorrespondence to Nicholas H. Wasserman . Rights and permissionsReprints and permissions About this articleWasserman, N.H. Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction. Can J Sci Math Techn 16 , 28–47 (2016). https://doi.org/10.1080/14926156.2015.1093200 Download citation Published : 01 January 2016 Issue Date : January 2016 DOI : https://doi.org/10.1080/14926156.2015.1093200 Share this articleAnyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 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Stack Exchange NetworkStack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Bachelor Thesis - Galois Theory Research Topics?I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was Galois theory. I am interested in doing 'my own' research, if you catch my drift. That is, I would like to apply the Galois theory I will be studying to something, and do some research. My professor mentioned some possible applications within coding-theory and cryptography. Do you have any specifics in mind? I would be happy to hear your insights. - abstract-algebra
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- 5 $\begingroup$ If you don't know Galois theory, that in itself sounds like a semester-long topic, worthy of understanding well whether or not you do any "research". You'll probably have more than enough on your plate learning Galois theory without trying to do something original (doubtful). $\endgroup$ – KCd Commented Jan 15, 2014 at 18:55
- 4 $\begingroup$ @KCd: To learn Galois theory and apply it meaningfully in some way that no one has done before : yes, that sounds like a tall order. But maybe the OP means that he wants to spend some time applying Galois theory himself rather than just reading about it, whether or not what he does has been done previously. (And if he doesn't mean this, perhaps he should...) $\endgroup$ – Pete L. Clark Commented Jan 15, 2014 at 19:05
- 4 $\begingroup$ I think it is easy to come up with polynomials whose Galois group hasn't been computed by hand so far - and it is very nice to see explicit radical expressions of the roots in the solvable case (for degree $8$, say). Of course this won't be any original research, but I would count this as research. $\endgroup$ – Martin Brandenburg Commented Jan 15, 2014 at 19:09
- $\begingroup$ @PeteL.Clark That is what I mean! I realize I won't be able to do original research at this point.. :-) $\endgroup$ – Numbersandsoon Commented Jan 15, 2014 at 19:12
- $\begingroup$ And it's very unlikely someone will expect you to, @BoSchmidt . I think Martin's idea is a very good one. Think of it. $\endgroup$ – DonAntonio Commented Jan 15, 2014 at 20:40
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Connections between Abstract Algebra and High School Algebra: A Few Connections Worth Exploringby Erin Baldinger, University of Minnesota; Shawn Broderick, Keene State College; Eileen Murray, Montclair State University; Nick Wasserman, Columbia University; and Diana White, Contributing Editor , University of Colorado Denver. Mathematicians often consider knowledge of how algebraic structure informs the nature of solving equations, simplifying expressions, and multiplying polynomials as crucial knowledge for a teacher to possess, and thus expect that all high school teachers have taken an introductory course in abstract algebra as part of a bachelor’s degree. This is far from reality, however, as many high school teachers do not have a degree in mathematics (or even mathematics education) and have pursued alternative pathways to meet content requirements of certification. Moreover, the mathematics education community knows that more mathematics preparation does not necessarily improve instruction (Darling-Hammond, 2000; Monk, 1994). In fact, some research has shown that more mathematics preparation may hinder a person’s ability to predict student difficulties with mathematics (Nathan & Petrosino, 2003; Nathan & Koedinger, 2000). Nevertheless, the requirements for traditional certification to teach secondary mathematics across the country continue to include an undergraduate major in the subject, and many mathematicians and mathematics educators still regard such advanced mathematics knowledge as potentially important for teachers. Given this, it is important that, as a field, we investigate the nature of the present mathematics content courses offered (and required) of prospective secondary mathematics teachers to gain a better understanding of which concepts and topics positively impact teachers’ instructional practice. That is, we need to explore links not just between abstract algebra and the content of secondary mathematics, but also to the teaching of that content (e.g., see Wasserman, 2015). In November 2015, a group of mathematicians and mathematics educators met as a working group around this topic at the annual meeting of the North American Chapter of the Psychology of Mathematics Education. We began to probe the impact understanding connections such as those described above might have on teachers’ instructional choices. For example, how does understanding the group axioms shift teacher instruction around solving equations? How does understanding integral domains shift teacher instruction around factoring? Through answering questions such as these, mathematicians and mathematics educators can better support teachers to connect advanced mathematical understanding to school mathematics in meaningful ways that enhance the quality of instruction. In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics. Example 1: Solving equations Solving equations and simplifying expressions is a technique used in multiple settings within mathematics. It uses the precise axioms of a group, but this is often not made transparent to students What would you do to solve this “one-step” equation? Many students are taught to subtract 5 from both sides to isolate the variable x , and they might write something like this (crossing out the 5s on the left hand side): However, on closer inspection, a variety of algebraic properties come to bear that the above work suppresses. (See Wasserman [2014] for a more complete elaboration and discussion.) An expanded version might look like this, with justifications for each step. ( x + 5) + -5 = (12) + -5 (Additive Equivalence) x + (5 + -5) = 12 + -5 (Associativity of addition) x + 0 = 12 + -5 (Additive Inverse) x = 12 + -5 (Identity Element for addition) x = 7 (Closure under addition) Similarly, if attention is given to algebraic properties used to solve equations, the solution to an equation of the form 5 x =12 might appear as follows: ⅕*(5 x )= ⅕*12 (Equivalence) (⅕*5) x = ⅕*12 (Associativity of multiplication) 1* x = ⅕*12 (Multiplicative Inverse) x = ⅕*12 (Identity Element for multiplication) x = 12/5 (Closure) These solution techniques can be related to students’ learning of matrix algebra in a course on linear algebra. Specifically, students learn, under appropriate conditions, to solve matrix equations of the form AX = B using these same steps. In each case above, the last four steps being used – the ones “hidden” from view in the one-step cancellation process – are the precise axioms for a group. In the first case, we’re working on the additive group of integers, in the second on the nonzero multiplicative group of rational numbers, and in the last under the group of n by n square matrices with nonzero determinant (i.e., invertible) under matrix multiplication. Thus, these are three a priori separate problems, all united by the same algebraic structure of a group – and that structure becomes evident in the algebraic solution process. Wasserman and Stockton (2013) discuss one vignette for how such knowledge might be incorporated into secondary instruction. Example 2: Simplifying expressions As a related example, consider the following two samples of student work: In each case, clearly a form of “cancellation” is being attempted. But what, technically, results in “cancellation”? And what remains after the cancellation is complete? Do sin and sin -1 make “1”? Is the “ x ” still an exponent? While we recognize this “cancellation” as attending to both the inverse elements and the meaning of the identity element in the group of invertible functions, these are subtle issues that are often not clear to students, and they are often taught in isolation, without the underlying structure being made apparent. In using the above two examples to illustrate, we do not intend to imply that teachers should require students to make explicit each and every use of a mathematical property when they solve equations. Rather, we aim to draw attention to the importance of recognizing the consistency going on across all of these examples of solving equations. Moreover, it is the collective power of individual properties – as they form the group (or ring/field) axioms – that allow for algebraic solution approaches and also help reconcile the meaning of “cancellation” in these different contexts as an interaction of both inverse and identity elements. Example 3: Polynomials and Factoring As another example of the connection between abstract algebra and secondary mathematics, we consider the problem of multiplying two polynomials. (See Baldinger [2013, 2014] for additional examples of this type.) In high school, students learn that the degree of the product of two nonzero polynomials is the sum of the degrees of the factors. Yet this does not hold in all types of algebraic settings. Consider, for example, the product of the following two polynomials when working modulo 7 versus modulo 8. As mathematicians, we of course recognize that the the degree of the product of two polynomials is the sum of the degrees of the factors — when the coefficients are elements of an integral domain, but that this relationship need not hold in other settings. Students, however, may be mystified when they first encounter an example like this in modular arithmetic, as their prior conceptions and understandings are being challenged, and they are thus being asked to deepen their understanding of the underlying structures that permit a result to hold in one setting, but break down in another. This example also ties directly into student misconceptions. For example, we teach students in high school that if the product of two polynomials is zero, then to solve we set each one separately equal to zero. Yet this does not hold with nonzero numbers. For example, working in polynomials with real coefficients, we know that f ( x ) * g( x )=0 implies either f ( x ) = 0 or g ( x ) = 0. Yet it is not the case that if f ( x ) * g ( x ) = 4, then either f ( x ) = 2 or g ( x ) = 2. The three above examples represent just a few of the many connections between abstract algebra and secondary mathematics. There has been a longstanding debate in the mathematics and mathematics education communities concerning the knowledge secondary mathematics teachers need to provide effective instruction. Central to this debate is what content knowledge secondary teachers should have in order to communicate mathematics to their students, assess student thinking, and make curricular and instructional decisions. This debate has already led to many fruitful projects (e.g., Connecting Middle School and College Mathematics [(CM) 2 ] (Papick, n.d.); Mathematics Education for Teachers I (2001) and II (2012); Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations (Heid, Wilson, & Blume, in press). A common thread in these projects is the belief that mathematics teachers should have a strong mathematical foundation along with the knowledge of how advanced mathematics is connected to secondary mathematics (Papick, 2011). But questions remain regarding what secondary content stems from connections to advanced mathematics, which connections are important, and how might knowledge of such connections influence practice. Our working group hopes to continue to explore these connections and contribute to our collective understanding of teacher education. Baldinger, E. (2013). Connecting abstract algebra to high school algebra. In Martinez, M. & Castro Superfine, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 733–736). Chicago, IL: University of Illinois at Chicago. Baldinger, E. (2014). Studying abstract algebra to teach high school algebra: Investigating future teachers’ development of mathematical knowledge for teaching (Unpublished doctoral dissertation). Stanford University, Stanford, CA. Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Issues in Mathematics Education, Vol. 11). Providence, RI: American Mathematical Society. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II (Issues in Mathematics Education, Vol. 17). Providence, RI: American Mathematical Society. Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1). Retrieved from http://epaa.asu.edu Heid, M. K., Wilson, P., & Blume, G. W. (in press). Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations. Charlotte, NC: Information Age Publishing. Monk, D. H. (1994). Subject matter preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145. Nathan, M. J. & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18, 209–237. Nathan, M. J. & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Education Research Journal, 40, 905–928. Papick, I. (n.d.) Connecting Middle School and College Mathematics Project. Retrieved March 7, 2015 from http://www.teachmathmissouri.org/ Papick, I. J. (2011). Strengthening the mathematical content knowledge of middle and secondary mathematics teachers. Notices of the AMS, 58(3), 389-392. Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K-12 mathematics teachers. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 24, 191–214. Wasserman, N. (2015). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education (online first). DOI: 10.1080/14926156.2015.1093200 Wasserman, N. & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), pp. 20–22. 1 Response to Connections between Abstract Algebra and High School Algebra: A Few Connections Worth ExploringAs a former IB student who took HL mathematics I was thankful to have experienced and learned about abstract algebra in high school. As a math major when I got to linear algebra it was so much easier for me than most of my classmates because I had that experience. Even when I got to modern algebra this year it was still not as difficult as people made it out to be because of that prior experience. However, watching my classmates struggle with those classes simply because they didn’t have the prior experience was frustrating to me. Most high school math classes don’t cover abstract algebra and it leaves the students who pursue mathematics in university at a disadvantage. I believe it should be a course anyone interested in pursuing mathematics in university should have access to in high school. It makes those university courses which are required to graduate with a math degree (at least at my university) much easier as well as the fact that it expands the mathematical framework a student has. This allows them to approach problems in different ways and find solutions in much easier and simpler ways. However, this would mean that schools would have to higher teachers who are math majors that want to teach instead of having teachers who pursue a career in education and get forced to teach math courses. So I feel like the best solution would be for some university to offer students who are interested in pursuing a career in or studying mathematics a course online that they can take to teach themselves the concepts of abstract algebra so they are more prepared in university. Comments are closed. 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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras.The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and ...
Algebra permeates all of our mathematical intuitions. In fact the first mathematical concepts we ever encounter are the foundation of the subject. Let me summarize the first six to seven years of your mathematical education: The concept of Unity. The number 1. You probably always understood this, even as a little baby. ↓
applications of abstract algebra. A basic knowledge of set theory, mathe-matical induction, equivalence relations, and matrices is a must. Even more important is the ability to read and understand mathematical proofs. In this chapter we will outline the background needed for a course in abstract algebra. 1.1 A Short Note on Proofs
Two tools of abstract algebra are. (a) axiomatic theories that define classes of algebras of data; and. (b) homomorphisms and isomorphisms that compare and classify the algebras. The achievement of abstract algebra is to allow the study of properties of calculations that are independent of the nature of the data and its representation.
Two interesting master thesis topics (related to group theory) that came to mind are: (1) The word problem in group theory: Given a group presented using generators and relations, can you tell whether a random element is actually equal to the identity? Surprisingly the answer is "No."
This article lays emphasis on how Abstract Algebra topic of Group Theory can be applied to the problems of Natural Language Processing. ... 39-87, 2015) as a theoretical framework, the thesis ...
each topic is organized as follows, although the order may vary slightly from topic to topic: 1Axioms are also sometimes called "laws". For example, you're probably familiar with the "commu-tative law" for addition, which says that a+ b= b+ a. But this isn't really a law, debated and approved ... Abstract Algebra — Lecture #1 1 ! ...
Abstract Algebra Thesis Topics - Free download as PDF File (.pdf), Text File (.txt) or read online for free. The document discusses the challenges of writing a thesis on abstract algebra, including selecting a topic, conducting thorough research, and organizing thoughts. It states that every step presents its own difficulties, which are intensified for abstract algebra.
This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a ...
10. I once wrote a detailed answer of some topics in geometric and combinatorial group theory which would be suitable for a talk or a master's thesis. This is the branch of group theory which deals with (loosely) actions of groups, presentations, a bit of algebraic geometry, etc. The post can be found here.
Microsoft Word - Suominen Dissertation.docx. ABSTRACT ALGEBRA AND SECONDARY SCHOOL MATHEMATICS: IDENTIFYING. AND CLASSIFYING MATHEMATICAL CONNECTIONS. By. Ashley Luan Suominen. (Under the Direction of AnnaMarie Conner) ABSTRACT. Many stakeholders concur that secondary teacher preparation programs should include.
Dissertation Topics in Abstract Algebra - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This document discusses writing a dissertation in abstract algebra and getting assistance. Abstract algebra deals with intricate concepts and rigorous proofs, making it challenging for students. The organization offers expert assistance from mathematicians and writers for ...
Here are some topics which I enjoy and thought that you might enjoy given your coursework. Mathematical Cryptography and the development of post quantum cryptosystems (heavy on number theory applications of abstract algebra and advanced linear algebra) Game Tree Theory (probability theory and algebraic manipulations).
General. Abstract algebra is the set of advanced topics in algebra that deal with abstract algebraic structures rather than the usual number systems. The zero set of a collection of polynomials. An algebraic variety is one of the the fundamental objects in algebraic geometry. A Boolean algebra is an algebra where the multiplication and addition ...
Thesis Topics in Abstract Algebra - Free download as PDF File (.pdf), Text File (.txt) or read online for free.
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics—and their progression across elementary, middle, and secondary mathematics— where teaching may be transformed by teachers' knowledge of abstract algebra are ...
3. I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was Galois theory. I am interested in doing 'my own' research, if you catch my drift. That is, I would like to apply the Galois theory I will be ...
In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics. Example 1: Solving equations. Solving equations and simplifying expressions is a technique used in multiple settings within mathematics. It uses the precise axioms of a group, but this ...
We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding. Chairs: George Bergman and Tony Feng.
Video (online) Consult the top 50 dissertations / theses for your research on the topic 'Algebra, Linear.'. Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard ...
Broadly, Christina described how the lessons prompted undergraduates to link advanced abstract algebra topics to secondary school. She said, "I think it gave motivation behind things and made them think through topics, to take that abstract stuff and hopefully, maybe, bring it down to a level where they could understand …before applying it ...