Abstract Algebra I
Purpose of Course showclose
The study of “abstract algebra” grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted. For example, we are familiar with the notion that real numbers are closed under multiplication and division (that is, if we add or multiply a real number, we get a real number). But if we divide one integer by another integer, we may not get an integer as a result—meaning that integers are not closed under division. We also know that if we take any two integers and multiply them in either order, we get the same result—a principle known as the commutative principle of multiplication for integers. By contrast, matrix multiplication is not generally commutative. Students of abstract algebra are interested in these sorts of properties, as they want to determine which properties hold true for any set of mathematical objects under certain operations and which types of structures result when we perform certain operations. Abstract algebra has applications in a variety of diverse fields, including computation, physics, and economics and, as a result, is an important area in mathematics.
We will begin this course by reviewing basic set theory, integers, and functions in order to understand how algebraic operations arise and are used. We then will proceed to the heart of the course, which is an exploration of the fundamentals of groups, rings, and fields.
Learning Outcomes showclose
 Describe and generate groups, rings, and fields.
 Relate abstract algebraic constructs to more familiar number sets and operations and see from where the constructs derive.
 Identify examples of specific constructs.
 Identify and differentiate between different structures and understand how changing properties give rise to new structures.
 Explain the theory behind relations and functions and identify domains and images of functions, based on the structures given.
 Explain how functions may relate seemingly dissimilar structures to each other and how knowing properties of one structure allows us to know the same properties in the related structure, if certain functions exist between them.
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plugins or software (e.g. Adobe Reader, QuickTime, and Flash viewer).
√ Be competent in the English language.
√ Have read the Saylor Student Handbook.
Unit Outline show close

Unit 1: Preliminary Information
Before we can begin looking at algebraic structures, such as groups, rings, and fields, we must review the information that provides the basis for those structures. All of these structures have sets, and set theory applies to groups at their core. Next, we will review the set of integers, as integers have all the properties that groups possess. Indeed, the set of integers, Z, with the operation “+” is itself a group.
Unit 1 Time Advisory show close
After discussing sets and operations on them, we will review relations and functions, as the concept of homomorphisms and isomorphisms arise from special cases of functions from one group, ring, or field to another. Finally, we will look at a special class of function known as the permutation, which plays a major role in group theory.
Unit 1 Learning Outcomes show close
 1.1 Basic Set Theory

1.1.1 Definition of Sets
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations” (PDF)
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Instructions: Please click on the link above. For the section on sets, read “1.2 Sets and Equivalence Relations,” pages 45. Please note that this PDF file will be used for the entire course. It will be referenced for readings and assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Reading: Reading: Wikibooks: “Set Theory/Sets”
Link: Wikibooks: “Set Theory/Sets” (PDF)
Instructions: Please read the entire webpage. This link provides considerable, condensed information on sets and set theory. For most, this will be a review of the primary terminology, axioms, and properties of sets.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikibooks version of this article here (HTML).  Lecture: Harvard University Extension: Dr. Paul Bamberg’s “Math E102 Sets, Counting, and Probability”: “Probability, Intuition, and Axioms”
Links: Harvard University Extension: Dr. Paul Bamberg’s “Math E102 Sets, Counting, and Probability”: “Probability, Intuition, and Axioms” (YouTube)
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Instructions: Please click on the link, and watch the video in its entirety. To choose another format, scroll down the webpage to Lecture 1 and choose the format most appropriate for your internet connection. The video mainly discusses probability, but the 45:30 mark has a useful discussion on set terminology and notation. Dr. Bamberg is a former Rhodes Scholar in theoretical physics and is now a fulltime math instructor at Harvard University’s Extension School’s Open Learning Initiative.
Terms of Use: This video has been reposted with permission for nonprofit educational use by Harvard University. The original version can be found here.  Lecture: Harvard University Extension: Dr. Paul Bamberg’s “Math E102 Sets, Counting, and Probability”: “Probability by Counting and InclusionExclusion”
Link: Harvard University Extension: Dr. Paul Bamberg’s “Math E102 Sets, Counting, and Probability”: “Probability by Counting and InclusionExclusion” (YouTube)
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Instructions: Please click on the link, and watch the video in its entirety. To choose another format, scroll down the webpage to Lecture 2 and choose the format most appropriate for your internet connection. The video clip discusses DeMorgan’s laws and gives an example of a Venn diagram. Dr. Bamberg is a former Rhodes Scholar in theoretical physics and is now a fulltime math instructor at Harvard University’s Extension School’s Open Learning Initiative.
Terms of Use: This video has been reposted with permission for nonprofit educational use by Harvard University. The original version can be found here.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 1 and 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 1 and 2” (PDF)
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Instructions: Work through problems 1 and 2 on page 18. After you have completed each problem, check your answers with the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”

1.1.2 Examples of Sets
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations” (PDF)
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Instructions: For the section on examples of sets, read “1.2 Sets and Equivalence Relations,” page 5. This PDF file will be used for the entire course. It will be referenced for readings and assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: Xoax.net’s “Algebra 1: Simple Sets”
Link: YouTube: Xoax.net's “Algebra 1: Simple Sets” (YouTube)
Instructions: Please click on the link, and view this brief video in its entirety (3:58 minutes) to learn about types of sets and ways to describe sets.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”

1.1.3 Set Operations
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations” (PDF)
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Instructions: For the section on examples of sets, read “1.2 Sets and Equivalence Relations,” pages 58. Please note that page 5 is a review of the information you read in subunit 1.2.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: APUS07’s “Intersection, Union, and Complement”
Link: YouTube: APUS07’s “Intersection, Union, and Complement” (YouTube)
Instructions: Please click on the link, and view the brief video lecture in its entirety (2:50 minutes) for a discussion on sets and common set operations.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 6, 10, and 14”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 6, 10, and 14” (PDF)
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Instructions: Please work through problems 6, 10, and 14 on page 19. After you have completed all of the assigned problems, check your answers against the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”
 1.2 Integers

1.2.1 Divisors
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers” (PDF)
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Instructions: For the section on examples of sets, read “2.2 The Division Algorithm,” on pages 2629.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 19 and 23”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 19 and 23” (PDF)
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Instructions: Complete problems 19 and 23 on page 33. Then, check your answers against the solutions on page 396.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers”

1.2.2 Primes
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers” (PDF)
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Instructions: For the section on examples of sets, read “2.2 The Division Algorithm,” pages 2930.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 27 and 29”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 27 and 29” (PDF)
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Instructions: Complete problems 27 and 29 on page 33. Then, check your answers against the solutions on page 396.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “The Integers”

1.2.3 Congruences and Integers Modulo
 Reading: Northern Illinois University: John A. Beachy’s and William D. Blair’s Abstract Algebra: “Integers”: “Congruences”
Link: Northern Illinois University: John A. Beachy’s and William D. Blair’s Abstract Algebra: “Integers”: “Congruences” (HTML)
Instructions: Please click the link above, and read the text in its entirety. The page defines and discusses modulo n. Dr. John A. Beachy is Professor Emeritus of Mathematical Sciences at Northern Illinois University.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Congruence mod n”
Link: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Congruence mod n” (YouTube)
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Instructions: Please click on the link, and watch the video. To choose a different format, scroll down to Lecture 6 (Week 2, Lecture 3) and choose the format most appropriate for your internet connection. The video clip ties integers mod n to cosets, which helps with the understanding of cosets later. Please watch the entire video (about 52 minutes).
Terms of Use: This video has been reposted with permission for nonprofit educational use by Harvard University. The original version can be found here.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problem 1”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problem 1” (PDF)
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Instructions: Complete problem 1 on page 49. Then, check your answer with the solution on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Northern Illinois University: John A. Beachy’s and William D. Blair’s Abstract Algebra: “Integers”: “Congruences”

1.3 Relations and Functions
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” “Sets and Equivalence Relations” (PDF)
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Instructions: For the section on examples of sets, read “1.2 Sets and Equivalence Relations,” pages 815.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: ClevelandMath’s “Relations and Functions”
Link: YouTube: ClevelandMath’s “Relations and Functions” (YouTube)
Instructions: Please click on the link, and view the video in its entirety (6:51 minutes). In this video lecture, Mr. Durbin discusses relations and functions, using various common notations. Since abstract algebra is largely about properties of sets with operators acting on them, this is a good introduction.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 18, 20, and 22”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 18, 20, and 22” (PDF)
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Instructions: Work through problems 18, 20, and 22 on page 19. Then, check your answers with the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Sets and Equivalence Relations”

1.4 Permutations
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Permutation”
Link: Wolfram MathWorld: Eric. W. Weisstein’s “Permutation” (HTML)
Instructions: Please click on the link to read the information on the entire webpage. This webpage contains a concise rendering of information on general permutations. For those who have had several classes in math, this represents a good refresher on the topic.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Lecture: YouTube: Burny1’s “Group Theory Permutations”
Link: YouTube: Burny1’s “Group Theory Permutations” (YouTube)
Instructions: Please click on the link, and view the brief video in its entirety (3:41 minutes). The video discusses general permutations and then discusses permutations in light of groups. This is a good introduction to the concept.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Permutation”

Unit 2: Fundamentals of Groups
Groups are the most fundamental of all algebraic structures. Consisting of a set with an operator (often called a binary operator), groups are simple, yet powerful, entities with applications in fields such as physics and economics. With fewer properties than, say, the set of real numbers with addition and multiplication, general groups give rise to structures that are as elegant as they are sometimes strange. As such, the study of groups is often where students of mathematics “trip up.” To avoid problems, do not make assumptions that are not available. For instance, we are aware that the field of real numbers is commutative under the property of multiplication. That is, for any real numbers a and b, a*b = b*a. However, while n?n matrices are groups under matrix multiplication, we can show examples where AB ? BA.
Unit 2 Time Advisory show close
We will then study some examples of groups, beginning with the finite group, which you will likely find easy to study. We will also look at some special types of groups, such as cyclic and permutation groups. Cyclic groups are generated from a single element. In fact, if the set G contained nothing but powers of some element g, then G = <g> = {g^{n }n is an integer} and g would be called the “generator of G.” Interestingly, a certain set of permutations of some set M is also a group (a permutation group). The set of all possible permutations on M is called the symmetric group of M. This is an important group that has relevance in Abstract Algebra II, when we will study Galois Theory. Another important example, especially in Linear Algebra, is the general linear group of invertible matrices. You will see more of this group in Abstract Algebra II.
After defining and examining a few examples of groups, we will see that any subset of a set with group properties for an operator is itself a group if it has the same properties. These groups are called “subgroups.” We will also consider functions from one group to another. Any function (also called a “mapping”) that retains consistent properties from one group to the next is called a homomorphism. That is, as we consider an operation * on group G and ? on group H, if some mapping f on G paired elements of G with those of H (that is, f(g) = h for some g in G and h in H) such that f(a*b) = f(a)?f(b), then f would be a homomorphism of G into H, and G and H would be considered homomorphic. If the mapping f were also 11 and onto (that is, the domain or image of the mapping covers all of H), f would be an isomorphism. The study of such mappings is important, for if H were a simpler group than G and we discovered that H and G were isomorphic, then anything we discover about H is also true of G.
We will end this unit with cosets. Cosets are formed of true subgroups of one group and single elements of the larger group. If all cosets are of the form gH = Hg, where H is a subgroup of G and g is in G, then H is called normal. This is important to remember because, in general, gH may not be equal to Hg. This is why we cannot assume commutativity. Cosets are analogous to equivalence classes on the integers, because they partition groups into distinct sets. Factor groups are normal subgroups.
Unit 2 Learning Outcomes show close

2.1 Definition of a Group
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” (PDF)
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Instructions: For the section on examples of sets, read “3.2 Definitions and Examples,” pages 4047.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: Nathan Carter's “Visual Group Theory”: “Part I” and “Visual Group Theory”: “Part II”
Link: YouTube: Nathan Carter's “Visual Group Theory”: “Part I” (YouTube) and “Visual Group Theory”: “Part II” (YouTube)
Instructions: Please click on the links above, and view each video in its entirety (1:28 minutes for “Part I” and 9:58 minutes for “Part II”). These video lecture is an interesting way of discussing groups in terms of visual symmetries.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 2, 8, 15, and 17”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 2, 8, 15, and 17” (PDF)
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Instructions: Work through problems 2, 8, 15, and 17 on page 51. After you complete each exercise, please see the solutions on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”
 2.2 Examples of Groups

2.2.1 Finite Groups
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Finite Group”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Finite Group” (HTML)
Instructions: Please click on the link to read the information on the webpage. This webpage contains a concise rendering of information on finite groups. It also contains a visual representation of several common finite groups.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: Finite Groups
Link: Wikipedia: “Finite Group” (PDF)
Instructions: Please read the entire web page. This webpage contains a concise rendering of information on finite groups. It also contains a visual representation of several common finite groups.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).  Web Media: YouTube: Klein Four's “Finite Simple Group (of Order 2)” Humor Video
Link: YouTube: Klein Four’s “Finite Simple Group (of Order 2)” (YouTube) Humor Video
Instructions: Please note that viewing this video is optional and this humorous video is meant to entertain. Who says mathematicians can't laugh? This is a nowfamous piece of group theory humor set to a fourpart a capella arrangement. Klein Four, a group of Northwestern University mathematicians, recorded this in front of an audience in their math department in November, 2006. Though intended strictly for humor, the use of terminology is correct and useful information can still be gleaned from the video. Please click on the link, and view the video in its entirety (about 3 minutes).
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Finite Group”

2.2.2 Cyclic Groups
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups” (PDF)
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Instructions: For the section on examples of sets, read Chapter 4, “Cyclic Groups”, pages 5773.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Reading: Knowledgerush: “Cyclic Group”
Link: Knowlegerush: “Cyclic Group” (PDF)
Instructions: Click on the link above, and read the webpage in its entirety for a discussion on cyclic groups and good examples of cyclic groups. The page contains information on properties of cyclic groups and ties cyclic groups to other group types that will be covered later.
Terms of Use: The material linked above is licensed under the GNU Free Documentation License (HTML). The original version can be found here (HTML).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”: “Exercise Problems 3 and 4”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”: “Exercise Problems 3 and 4” (PDF)
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Instructions: Complete problems 3 and 4 on page 69. After you have finished each problem, check the solutions on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributid to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”

2.2.3 Permutation Groups
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups” (PDF)
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Instructions: For the section on examples of sets, read Chapter 5 “Permutation Groups”, pages 7491.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: “Symmetries of a star and Its Permutation Group”
Link: YouTube: “Symmetries of Star and Its Permutation Group” (YouTube)
Instructions: Please click on the link, and view the video in its entirety. The video discusses a permutation group of a regular geometric figure (a Star of David). This is an interesting visual presentation.
Terms of Use: The linked material above has been reposted by the kind permission of Youtube User: S22105 and can be viewed in its original form here. Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”: “Exercise Problems 1, 2, and 3”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”: “Exercise Problems 1, 2, and 3” (PDF)
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Instructions: Work through problems 1, 2, and 3 on page 88. After you complete these, check your answers on page 398.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”

2.2.4 Symmetric Group
 Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Symmetric Group”
Link: Wolfram MathWorld: Eric. W. Weisstein’s “Symmetric Group” (HTML)
Instructions: Please click on the link to read the information on the webpage. This webpage contains a concise rendering of information on symmetric groups. It also contains a visual representation of symmetric group multiplication table.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “Symmetric Group”
Link: Wikipedia: “Symmetric Group” (PDF)
Instructions: Please read the entire webpage. This webpage contains a concise rendering of information on finite groups. It also contains a visual representation of several common finite groups.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Symmetric Group”

2.2.5 General Linear Group of Invertible Matrices
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “General Linear Group”
Link: Wolfram MathWorld: Eric W. Weisstein’s “General Linear Group” (HTML)
Instructions: Please click on the link to read the information on the webpage for a concise rendering of information on the general linear group. It is short but has a number of links to related topics of interest.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “General Linear Group”
Link: Wikipedia: “General Linear Group” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on general linear groups, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “General Linear Group”

2.3 Subgroups
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” (PDF)
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Instructions: For the section on examples of sets, read “3.3 Subgroups,” pages 4750.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free (HTML). It is attributed to Thomas W. Judson and the original version can be found Documentation Licensehere (PDF).  Lecture: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 8” Discussion
Link: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 8 ”Discussion (YouTube)
Instructions: Please click on the link, and view the video in its entirety (5:42 minutes) to learn about subgroups.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 34 and 40”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 34 and 40” (PDF)
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Instructions: Complete problems 34 and 40 on page 52. Then, check your answers against the solutions on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”

2.4 Homomorphisms
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms” (PDF)
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Instructions: For the section on examples of sets, read “Chapter 11: Homomorphisms,” pages 165171.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 5” Discussion
Link: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 5” Discussion (YouTube)
Instructions: Please click on the link, and view the video in its entirety (7:39 minutes) for a discussion on group homomorphisms and isomorphisms in a fairly straightforward manner.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”: “Exercise Problems 2, 4, and 9”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”: “Exercise Problems 2, 4, and 9” (PDF)
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Instructions: Try to do problems 2, 4, and 9 on page 173. After you complete the assigned problems, check the solutions on page 402.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”

2.5 Isomorphisms
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms” (PDF)
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Instructions: For the section on examples of sets, read “Chapter 9: Isomorphisms,” pages 141151.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”: “Exercise Problems 1, 2, 6, and 8”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”: “Exercise Problems 1, 2, 6, and 8” (PDF)
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Instructions: Work on problems 1, 2, 6, and 8 on page 151. Then, check your answers against the solutions on page 401.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”
 2.6 Cosets, Normal Subgroups, and Factor Groups

2.6.1 Cosets
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and LaGrange’s Theorem”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and LaGrange’s Theorem” (PDF)
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Instructions: For the section on examples of sets, read “6.1 Cosets,” pages 9294.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). The original version can be found here (PDF).  Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Equivalence Relations; Cosets; Examples”
Link: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Equivalence Relations; Cosets; Examples” (YouTube)
Also Available in: Adobe Flash, Quicktime, or Mp3
Instructions: Please click on the link, and watch the video in its entirety. To choose another format, scroll down to Week 2 and choose the format most appropriate for your internet connection to download the second lecture listed in Week 2 titled “Equivalence Relations; Cosets; Examples.” Please watch the entire video (about 47 minutes) to learn about equivalence relations and how they are related to cosets.
Terms of Use: This video has been reposted with permission for nonprofit educational use by Harvard University. The original version can be found here.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and Lagrange's Theorem”: “Exercise Problems 1 and 5”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and Lagrange's Theorem”: “Exercise Problems 1 and 5” (PDF)
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Instructions: Try to do problems 1 and 5 on page 98. Then, check the solutions on page 399.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and LaGrange’s Theorem”

2.6.2 Normal Subgroups and Factor Groups
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups” (PDF)
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Instructions: For the section on examples of sets, read “Chapter 10: Normal Subgroups and Factor Groups,” pages 155161.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: YouTube: “Normal Subgroup Example 1”
Link: YouTube: “Normal Subgroup Example 1” (YouTube)
Instructions: Please click on the link, and view the short video, which is approximately 2 minutes, for an example of a normal subgroup.
Terms of Use: The linked material above has been reposted by the kind permission of Youtube User: S22105, and can be viewed in its original form here. Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”: “Exercise Problems 1, 8, and 13”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”: “Exercise Problems 1, 8, and 13” (PDF)
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Instructions: Work through problems 1, 8, and 13 on page 163. Then, check the solutions on page 402.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”

Unit 3: Fundamentals of Rings
Rings are “larger” structures than groups, because they are sets with two binary operators instead of one and because they have more properties than groups. Technically, a ring is a commutative group, under addition, with the added property of being associative under multiplication. The distributive property connects the two operations. The study of rings began as a study of generalizations on addition and multiplication of integers. Interestingly, the desire to generalize structures of number sets and operations came from the interest in solving Fermat’s Last Theorem.
Unit 3 Time Advisory show close
We begin the unit with a more formal definition of rings and look at ring properties. We will then look at commutative and noncommutative rings (note that if a ring is commutative under multiplication, it is defined as a “commutative ring”). The study of these two classes of rings arose independently and mathematicians tended not to work on both at the same time. Though most research has focused on commutative rings, noncommutative rings have become an important area of study over the last sixty years. We do wish to familiarize ourselves with noncommutative rings, but we will not spend too much time studying them, for such a study could realistically command an entire course of its own.
Next, we will move into the concept of ideals, which is based on the work of J. W. R. Dedekind. It was Dedekind and his ideals that first moved abstract algebra from number theory to structure theory. Dedekind was looking for the properties of “ideal complex numbers,” but he generalized his findings to abstract structures rather than working strictly with number sets. He defined an ideal as a subring whose elements, when multiplied by any element of the larger ring, result in an element of the subring. That is, for a ring R and a subring S, if the product rs is in S for every r in R and s in S, then S is an ideal. A maximal ideal is the “largest proper ideal” of a ring. That is, if R is a ring and I is an ideal of R, then I is maximal, if there are no largerproper ideals in R. Another way of approaching maximal ideals is if J is any ideal in R containing I, then either J = R or J = I. Prime ideals extend the idea of prime numbers to rings. That is, rather than looking at properties of prime numbers, we can look at subsets of a ring that share many of the properties of prime numbers. This enables us to look at mathematical structures other than the familiar ones that may share similar qualities.
Modules over rings, for instance, are generalizations of vector spaces. Vector spaces have scalars that form fields. Modules only require scalars to form rings, which are simpler and may not have an identity element over module multiplication. The primary difference between modules and vector spaces are that a module does not necessarily have a basis and certain modules may not have unique rank.
As with groups, homomorphisms are mappings from one ring to another that preserve structures and isomorphisms are homomorphisms that are also onto and 11. As with the study of mappings on groups, the interest is in seeing what kinds of ring structures arise when we have mappings across rings that may not be entirely similar.
We will end the unit looking at a particular kind of ring, called the polynomial ring. These are formed by polynomials in one or more variables with coefficients in some ring R. Again, the interest is in finding generalizations of what we know about familiar polynomials with number coefficients. What we will learn is that polynomial rings are either prime or factor uniquely.
Unit 3 Learning Outcomes show close

3.1 Definitions and Properties of Rings
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)
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Instructions: For the section on examples of sets, read “16.1 Rings,”pages 239244.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Rings”
Link: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Rings Part 1,” "Rings Part 2," and "Rings Part 3" (YouTube)
Also Available in: Adobe Flash, Quicktime, or Mp3
Instructions: Please click on the links and watch each video in its entirety. To choose another format, scroll down to Week 9 and choose the format most appropriate for your internet connection. Then download Parts 13 of the “Rings: Examples of Rings and Basic Properties and Constructions” lectures listed in Week 9. Please watch all of the videos in their entirety (about 53 minutes for Part 1, about 48 minutes for Part 2, and about 50 minutes for Part 3). These video clips discuss properties and constructions of rings.
Terms of Use: This video has been posted with permission for nonprofit educational use by Harvard University. The original version can be found here.  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problems 1 and 3”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications:“Rings”: “Exercise Problems 1 and 3” (PDF)
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Instructions: Try to do problems 1 and 3 on page 257. Then, check the solutions on page 405.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”

3.2 Example of Rings
 Reading: Wikipedia: “Ring (Mathematics)”
Link: Wikipedia: “Ring (Mathematics)” (PDF)
Instructions: Please read the entire webpage. This webpage contains a concise rendering of information on rings, including useful examples.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).  Reading: Wolfram MathWorld: Eric W. Weisstein’s “Ring”
Link: Wolfram MathWorld: Eric. W. Weisstein’s “Ring” (HTML)
Instructions: Please click on the link to read the information on the webpage for a concise rendering of information on rings. It is short but has a number of links to related topics of interest. Halfway down the page there is an interesting explanation of optional properties of rings and examples of rings that rise from changes in properties.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.
 Reading: Wikipedia: “Ring (Mathematics)”
 3.3 Commutative and NonCommutative Rings

3.3.1 Commutative Rings
 Reading: Wikipedia: Commutative Ring
Link: Wikipedia: “Commutative Ring” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on commutative rings, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wikipedia: Commutative Ring

3.3.2 NonCommutative Rings
 Reading: Wikipedia: “Noncommutative Ring”
Link: Wikipedia: “Noncommutative Ring” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on noncommutative rings, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wikipedia: “Noncommutative Ring”

3.4 Ideals
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)
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Instructions: For the section on examples of sets, read “16.3 Ring Homomorphisms and Ideals,” pages 246250.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 5”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings:” “Exercise Problem 5” (PDF)
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Instructions: Practice what you have learned by completing problem 5 on page 258. Then, check the solution on page 404.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”

3.5 Maximal and Prime Ideals
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)
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Instructions: For the section on examples of sets, read “16.4 Maximal and Prime Ideals,” pages 250252.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 4”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 4” (PDF)
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Instructions: Try to do problem 4 on page 258. Then, check the solution on page 404.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”

3.6 Modules
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Module”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Module” (HTML)
Instructions: Please click on the link to read the information on the page. This webpage contains a concise rendering of information on modules, which are abstractions of vector spaces, with the primary exception being the coefficients are over rings instead of fields. An example is the set of integers, which is a module over itself.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “Module (Mathematics)”
Link: Wikipedia: “Module (Mathematics)” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on modules, including various examples.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Module”

3.7 Ring Homomorphisms
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)
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Instructions: For the section on examples of sets, read “16.3 Ring Homomorphisms and Ideals,” pages 246249.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 19”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 19” (PDF)
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Instructions: Try to do problem 19 on page 259. Then, check the solution on page 404.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”

3.8 Ring Isomorphisms
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)
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Instructions: For the section on examples of sets, read “16.3 Ring Homomorphisms and Ideals,” pages 249250.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”

3.9 Polynomial Rings
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials” (PDF)
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Instructions: For the section on examples of sets, read “Chapter 17: Polynomials,” pages 263277.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials”: “Exercise Problem 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials”: “Exercise Problem 2” (PDF)
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Instructions: Try to do problem 2 on page 278, and then check the solution on page 405.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials”

Unit 4: Fundamentals of Fields
We conclude this course with a unit on fields. Fields are general sets with some defined operations of addition and multiplication that have all the familiar properties of rational, real, and complex numbers. They form commutative groups over both addition and multiplication and have the distributive property that connects the two operations.
Unit 4 Time Advisory show close
We will begin the unit with a formal definition of fields and consider field properties. We will then look at some examples of fields and learn about field extensions. An extension of some field K is nothing more than a field M that contains K as a subfield. Consider the real numbers R as an extension of the rational numbers Q. R contains irrational numbers and, more importantly, transcendental numbers. Transcendental numbers can never be solutions to algebraic equations. Further, if there are algebraic and transcendental extensions of fields in specific, what other kinds of properties might be missing from one field F but included in some minimal field containing F, in general?
For instance, there are splitting fields. If P(X) is a polynomial over some field K, then any field L containing K which P(X) can “split” into linear factors X  a_{i} is called a splitting field of P(X). If the field is the rational numbers Q and we consider a polynomial P(X) = X^{3} – 7 with coefficients in the rationals, one solution is irrational and two are complex, meaning that we would need an extension with those solutions, at minimum. Any extension that contains all three roots would be a splitting field.
There is also the issue of Pclosures. Suppose a field K does not have a certain property P. Then any extension L containing K that had property P would be called a Pclosure for K. One example is complex numbers being the algebraic closure for real numbers, since some polynomials with real coefficients have complex roots. In addition, an extension field of K is a separable closure for K if it contains the smallest set of all possible finite separable extensions of K within the algebraic closure.
Unit 4 Learning Outcomes show close

4.1 Definition and Properties of a Field
 Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Field” and “Field Axioms”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Field” (HTML) and “Field Axioms” (HTML)
Instructions: Please click on each link to read the information on each webpage. The first webpage titled “Field” contains a concise rendering of information on fields. It is very short, but provides a good explanation in a few sentences of what a field is and is not. The second webpage titled “Field Axioms” contains a concise rendering of information on field axioms (properties). It is very short, but provides a good explanation in a few sentences of what makes a field a field.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “Field (Mathematics)”
Link: Wikipedia: “Field (Mathematics)” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on fields, including various interesting examples.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Field” and “Field Axioms”

4.2 Extension Fields
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields” and “Rings”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields” and “Rings” (PDF)
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Instructions: Please read “21.1 Extension Fields,” pages 329340, and then read “16.3 Ring Homomorphisms and Ideals,” pages 246250.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 2” (PDF)
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Instructions: Work through problem 2 on page 350. Then, check the solution on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields” and “Rings”

4.3 Splitting Fields
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields” (PDF)
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Instructions: Please read “21.2 Splitting Fields,” pages 340343.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).  Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 3”
Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”: “Exercise Problem 3” (PDF)
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Instructions: Complete problem 3 on page 350, and then check your answer on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
 Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Fields”

4.4 Algebraic Closures
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Algebraic Closure”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Algebraic Closure” (HTML)
Instructions: Please click on the link to read the information on the webpage for a summary of algebraic closure. It is very short but provides a good explanation in a few sentences.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “Algebraic Closure”
Link: Wikipedia: “Algebraic Closure” (PDF)
Instructions: Please read the entire webpage. This webpage contains a concise rendering of information on algebraic closure, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Algebraic Closure”

4.5 Separability
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Separable Extension”
Link: Wolfram MathWorld: Eric W. Weisstein’s “Separable Extension” (HTML)
Instructions: Please click on the link to read the information on the webpage for information on separable field extensions. It is very short but provides a good explanation in a few sentences.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.  Reading: Wikipedia: “Separable Extensions”
Link: Wikipedia: “Separable Extension” (PDF)
Instructions: Please read the entire webpage for a concise rendering of information on separable field extensions, including various examples. This topic is useful toward Abstract Algebra II.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
 Reading: Wolfram MathWorld: Eric W. Weisstein’s “Separable Extension”

Final Exam
 Final Exam: The Saylor Foundation's MA231 Final Exam
Link: The Saylor Foundation's MA231 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation's MA231 Final Exam