Applied Abstract Algebra

Syllabus

Calendar

  • …
  • 4/16/2025: We went through the Groups and Homomorphism Slides, you should note that many of the properties of group homomorphisms are identical to properties of ring homomorphisms which we discussed before. At this point we have finished the material from chapter 7. Next week we will go through Cosets and Lagrange’s Theorem.
  • 4/9/2025: We spent much of the class working through the packet on motions of the triangle % square and on permutations. We then went through more of the Groups and Subgroups Slides; please go through the last section of that on your own. We have covered content from sections 7.1, 7.2, 7.3, and a bit of 7.5. We will look at at the Groups and Homomorphism Slides next class.
  • 4/2/2025: We started looking at groups, the first two sections of the Groups and Subgroups Slides. This matches roughly sections 7.1 and 7.2 in the text. We will work through the rest of the slide deck and start looking at the Groups and Homomorphism Slides.
  • 3/19/2025: Exam 1
  • 3/12/2025: Geometric Constructions
  • 3/5/2025 – We spent tonight reviewing additional examples from lat weeks material. Since we didn’t get to discuss any of the applications of rings this week we are moving the Rings exam to after break.
  • 2/26/2025: Tonight we covered material on maximal ideals, irreducible polynomials, quotient rings, and fields. We ended by showing that is \(p(x)\) is an irreducible polynomial in \(\mathbb{Z}_2[x]\), then \(\mathbb{Z}_2/(p(x))\) is a field and we can use this to define addition and multiplication of ordered tuples of bits in a computer algorithm. We have now finished what we will cover in class from chapters 1 through 6. Over the next two weeks we will cover a variety of topics from chapters 11, 13, 14, and 15.
  • 2/19/2025: We discussed subrings, ideals, and kernels tonight. Then we proved the First Isomorphism Theorem for Rings. We also looked at examples that will allow us to relate fields, maximal ideals, and irreducible polynomials. You should continue working on the exercises listed so far and looking over chapters 5 & 6.
  • 2/12/2025: We discussed homomorphisms and isomorphisms. In particular we reviewed the idea of linear transformations, row spaces, column spaces and null spaces from linear algebra. Then we defined and looked at properties of homomorphisms and isomorphisms. You should be familiar with these including domain, codomain, range/image, kernel, one-to-one/injective, and onto/surjective. Of particular interest are the evaluation maps from rings to polynomials to other rings, eg. \(\phi_a: R[x]\rightarrow S\) defined by \(\phi_a(f)=f(a)\). At this point we have finished chapter 3 and covered most of the general content in chapter 4. You can also skim through the content in chapters 5 and 6 for upcoming classes.
  • 2/5/2025: Tonight we looked very briefly at how we can perform basic arithmetic on curves of the form \(y^2=x^3+Ax+B\) which we will look at later in the semester when we discuss groups. We then covered the basic definitions of Rings, Integral Domains, and Fields. We looked at how many properties we take for granted such as \(\forall a\ a\times0=0\times a=0\), \(-(-a)=a\), \(-a\times b=-(a\times b\), and \(a\times b=a\times c\) are emergent properties and do not need to be assumed as part of our definitions. Finally, we saw that all finite integral domains are in fact fields and hinted at the idea that all finite fields have prime power order, i.e. the number of elements in the field equals \(p^k\) for some prime \(p\) and natural number \(k\). You should continue to look through the book, particularly chapter 3 Sections 3.1 and 3.2.
  • 1/29/2025: We went through the Review of MAT 207: Proofs type Material Slides tonight which reviewed and expanded on much of the material we covered last week. We then spent a little time discussing/recalling properties of polynomials. In particular we discussed the Factor Theorem the Remainder Theorem, and the Fundamental Theorem of Algebra. Connected to these we discussed polynomial division, the Euclidean Algorithm for polynomials and the idea that how/if we are able to factor a polynomial depends on the set of numbers we are looking at.
  • 1/22/2025: We spent most of our time looking at material on number theory and modular arithmetic. Then a little time discussing matrix arithmetic. One of the things we saw in both situations was that while there were always additive inverses, there were not always multiplicative inverses. In fact in both situations we could have \(a\times b=0\) with \(a,b\neq 0\), in which case we call \(a\) and \(b\) zero divisors. You should read through chapters 1 and 2 for next week.

Ring Theory

Assignments:

Complete one problem per section. Turn them in as you finish them, try not to let it drag out.

  1. Section 2.1: 14, 18&21 together, 22
  2. Section 2.3: 13, 14
  3. Section 3.1: 26, 30, 36
  4. Section 3.2: 28, 38, 40, 42
  5. Section 3.3: 20, 24
  6. Section 4.1: 14, 18, 20
  7. Section 4.2: 12, 14, 16
  8. Section 4.3: 16, 20
  9. Section 4.4: 12, 14 (Generalization of the SAT question), 16, 18
  10. Section 5.1: 10 & 11 together, 8 & 12 together
  11. Section 5.2: 8 & 11 together, 14
  12. Section 5.3: 4, 6, 10
  13. Section 6.1: 26, 34, 36
  14. Section 6.2: 10, 16, 22
  15. Section 6.3: 10, 16
  16. Section 11.6: 8,10,12,14,15,18
  17. Section 13: 2,6 (15pt problem)
  18. Section 14.1: 10,14, 19&20 together (15pt problem)
  19. Section 14.3: 4,6

Exam Guide

  • Readings: Selections from Chapter 1 through 6, 11, 13, 14, 15
  • Vocabulary: rings, zero divisors, units, integral domains, fields, ideals, cosets, quotient rings, irreducible polynomials, field extensions, splitting fields, maximal ideals, homomorphisms, isomorphisms, and kernels
  • Theorems: First Isomorphism Theorem for Rings, The two step subring test
  • Skills: demonstrate basic proficiency/ understanding of modular arithmetic, fill in addition and multiplication tables for finite rings, perform arithmetic with polynomials over fields (finite and infinite), carrying out arithmetic on bytes of information and the explain the connection to quotient rings. In general be able to demonstrate understanding of rings and fields and their basic properties. Be sure to look at the (A) type exercises in the text.
  • Concepts: connection between field extensions, quotient rings, and maximal ideals, limitations on geometric constructions with straightedge and compass in the context of field extensions

Group Theory

Assignments

Complete one problem per section. Turn them in as you finish them, try not to let it drag out.

  1. Section 7.1: 16, 18, 30
  2. Section 7.2: 24, 26, 30
  3. Section 7.3: 40, 42, 44, 56 & 57 (as one)
  4. Section 7.4: 26, 30, 32, 36
  5. Section 7.5: 18, 22, 24, 36
  6. …

Exam Guide

  • Readings: …
  • Vocabulary: groups, subgroups, orders of groups, orders of elements, permutations, homomorphisms, isomorphisms, endomorphisms, automorphisms, kernels, normal subgroups, quotient groups, cosets, Sylow \(p\)-subgroups, …
  • Theorems: First Isomorphism Theorem, Lagrange’s Theorem, Cayley’s Theorem, and Cauchy’s Theorem
  • Skills: Look at the following (A) type questions
    • Section 7.1: 4, 7, 10, 12
    • Section 7.2: 1, 9, 13, 15, 18
    • Section 7.3: 11, 12, 15, 19, 21, 26, 33
    • Section 7.4: 4-12, 16, 19, 23
    • Section 7.5: 3, 4, 6, 8, 9, 12
    • …
  • Concepts: key points in Algebraic Coding Theory and how they are connected to groups and to matrix operations, …
  • …

Old Slide Decks

Misc. Links and Handouts