Applied Abstract Algebra

Syllabus

Calendar

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  • 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. More Options to Come

Exam Guide

  • Readings: Selections from Chapter 1 through 6, 11, 13, 14, 15
  • Vocabulary: rings, zero divisors, units, integral domains, fields, quotient rings, irreducible polynomials, field, extensions, ideals, homomorphisms, isomorphisms, and kernels, …
  • Theorems: First Isomorphism Theorem for Rings, …
  • Skills: carrying out arithmetic on bytes of information and the connection to quotient rings,
    arithmetic over finite fields and its use in encryption schemes, …
  • Concepts: connection between field extensions, quotient rings, and maximal ideals, limitations on geometric constructions with straightedge and compass in the context of field extensions, …
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Group Theory

Assignments

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Exam Guide

  • Readings: …
  • Vocabulary: groups, subgroups, normal subgroups, Sylow \(p\)-subgroups, quotient groups, cosets, permutations, homomorphisms, isomorphisms, and kernels, …
  • Theorems: First Isomorphism Theorem, Lagrange’s Theorem, Cayley’s Theorem, and Cauchy’s Theorem
  • Skills: uses of groups and permutations in various encryption schemes, …
  • Concepts: key points in Algebraic Coding Theory and how they are connected to groups and to matrix operations, …
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Old Slide Decks

Misc. Links and Handouts