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Homework Solutions

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Lecture: Tuesday, Thursday 1:30-2:45, in Keller 402
Instructor: Ralph Freese
      Office: 305 Keller
      email:email me
      Office hours: T-Th 2:45-3:10 and by appointment (or just come to my office
                               and see if I'm in)

• Project Paper and Presentation. Besides the web and wikipedia in particular, a good source is Keith Conway's page.
  
  
  • Sample paper on cyclotomic polynomials:
    • SamplePaper-Cyclotomic.pdf.
    • SamplePaper-Cyclotomic.tex.
    • AlgebraRefs.bib.
  
  
  • Finite Fields: This project should involve two or maybe three people working together. The first part should cover the basic properties (the book covers these pretty well). The second should talk about actually constructing finite fields. Here are some slides that have a good explanation: Finite Fields.

    Conway Polynomials should also be discussed. The wikipedia page is pretty good. Also Frank Luebeck page does a nice job and actually gives many of them.

  • Topics in Ring Theory: For this project you could choose something from ring theory, possibly something from Chapter 10. Or you could cover Gaussian integers and Eisenstein integers. Gaussian integers is just the ring \(\mathbb Z[i] = \{a + bi : a, b \in \mathbb Z\}\). While Eisenstein integers are the ring \(E = \{a + b\omega : a, b \in \mathbb Z\}\), where \(\omega = (-1 + \sqrt 3i)/2\). Both have a norm which helps in understanding the arithmetic of the rings. For Gauassian integers it is (\(N(a+ib) = a^2 + b^2\) which is positive-definite. But the norm for Eisenstein is not positive-definite, which makes things interesting. Here is a sample paper with questions you can use to get started.
    • SamplePaper-GaussAndEisenstein.pdf.
    • SamplePaper-GaussAndEisenstein.tex.
    • AlgebraRefs.bib.
  
  
  • Straight-edge and compass constructions: Basically a length \(a\) can be constucted (from a unit length) iff \([\mathbb Q(a):\mathbb Q] = 2^k \) for some \(k\). You should outline the proof of this and use to show that doubling a cube, trisecting an angle, and squaring the circle are all impossible. You should also look into which regular polygons can be constructed. At least show a pentagon can be constructed and a septagon cannot. The key is to find the minimum polynomial of \(\zeta_n + \zeta_n^{-1}\), where \( \zeta_n = e^{2\pi i/n} \) defined in the cyctomic paper above.
  
  
• Group Action and Sylow's Theorem: Group Actions.


• Book: T. Hungerford, Abstract Algebra: An Introduction, third edition.
• Read Appendix A and Professor Yashinski's Notes on writing proofs.


• Other Good Books:
  • Herstein, Topics in Algebra.
  
  
• \({\rm \TeX}\) (and \({\rm \LaTeX}\)): is a software system for typing mathematics. You will be required to use this for the homework.
  • Instructions for installing and using it. After you have installed it I suggest making a directory (directories are called folders by Mac people) for our class. Something like Math412. Put exerssolns.sty in your new directory and put HW-1-YourName.tex in that directory. (If you prefer, you can put exerssolns.sty in ~/texmf/tex/latex or maybe ~/Library/texmf/tex/latex on a Mac.) Open TeXworks (or TeXshop on the Mac).
  • Read part of the book A not so short indtroduction to TeX. Chapter 3 talks about typesetting math and gives list for all the symbols.
  • The UH Library's \({\rm \LaTeX}\) pages.
  • You can run \({\rm \TeX}\) in the cloud using Overleaf.com.
  
  
• Homework: Download the first two files and change the file to have your name. In the file change "Billy Bob" to your name. The starred problems are to be written up very carefully, using complete sentences. (Afterall this is a writing intensive course.)
  • HW-1-YourName.tex
  • exerssolns.sty
  • pdf
  Future assignments will be on the Homework page.