Author Topic: Learning and Understanding in Abstract Algebra  (Read 338 times)

0 Members and 1 Guest are viewing this topic.

Nation of One

  • { }
  • { ∅, { ∅ } }
  • Posts: 4756
  • Life teaches me not to want it.
    • What Now?
Learning and Understanding in Abstract Algebra
« on: December 23, 2021, 11:26:47 pm »
Learning and Understanding in Abstract Algebra by Bradford R. Findell

It's really quite a table of contents for a dissertation paper, no?

Also:  Innovations in Teaching Abstract Algebra

https://www.d.umn.edu/~jgallian/
« Last Edit: December 24, 2021, 01:11:11 am by Half-Crazy Nobody »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Share on Facebook Share on Twitter


Nation of One

  • { }
  • { ∅, { ∅ } }
  • Posts: 4756
  • Life teaches me not to want it.
    • What Now?
Re: Learning and Understanding in Abstract Algebra
« Reply #1 on: January 19, 2022, 08:05:20 pm »
There has been a Coming Around Full Circle for me ...

I have decided to focus primarily on the Gallian text, "Contemporary Abstract Algebra" if only for the wealth of resources supporting the text available online. 

The whole structure of number theory rests on a single foundation, namely the algorithm for finding the greatest common divisor.  The Gallian text begins Chapter 0: Preliminaries with the Division Algorithm; hence, the process of finding greatest common divisor of a and n, gcd(a, n).

There is some Python code of interest hiding in our Murderous Resentment thread.  There is some C++ code to solve congruences left in the Programming as Mathematics thread.

In the University of One thread, there are Python programs for implementing Euclid's algorithm for computing GCD and extended GCD.

Also, the Display-Extended-Euclidean-Algorithm code on Github.

Abstract Algebra with GAP.

Links to Rainbolt/Gallian Lab Manual  for Contemporary Abstract Algebra.

Human Guidance with Abstract Algebra

misc:

Be forewarned that I find Paul Garrett's presentation of "Completing the Square" in the very beginning of "Intro to Abstract Algebra" to be confusing; and that the full course notes for Abstract Algebra are meant for graduate students majoring in mathematics.   I leave it here as a novel curiosity.

For, if one investigates the very first topic treated in the full "Abstract Algebra" notes, namely, "unique factorization," I find the notation more meaningful.  The Division Algorithm is the first topic treated in the Gallian text.

I like Paul Garrett's use of the term reduction modulo (as opposed to the standard "remainder").   Myself, I will only use it as an exotic reference.  I may print the first few pages, just to have a novel presentation of the foundational algorithm upon which the entire structure of the theory of numbers depends; that is, the algorithm for finding the greatest common divisor of two integers.

I may reflect upon different presentations such as this, and then call attention to them within my current Math Modules of Gallian's Contemporary Abstract Algebra.  These may supplement the Gallian text, and thereby enhance the presentation (personalized by my own research), enriching rather than confusing, even if the series extends into 100 notebooks.

On the other hand, since I do intend to explore several diverse sources, it might be better to devote a separate series of notebooks for "abstract algebra and the theory of numbers."   Then there is less chance of confusing the reader (myself).  Since Garrett assigns more meaningful variables, it might be more clear if I keep them separate from notes on Gallian text [CAA]; and then, from within those [CAA] modules, direct the reader/student/explorer to the relevant notebook in the generic abstract algebra and the theory of numbers series.

I am very tempted to insert such "novel representations" found in obscure places directly into the notebook I am using to "work through" Gallian's [CAA] text.   This would really personalize it for me at each step, with each theorem and definition.

For instance, Garrett's use of the term "reduction modulo" forced me to research elsewhere:

If you divide an integer a by a non-zero integer d, you get a = q×d + r where q is the quotient, d is the divisor and r is the remainder.

There are d possible remainders: 0, 1, 2, ... , d − 1.

The reduction modulo d of an integer is, loosely speaking, its remainder in the division by d.
__________________________________________________________
Reduction Modulo m:  Once a set of representatives has been chosen for the elements of Zm, we will call “r reduced modulo m”, written “r mod m”, the chosen representative for the class of r.
_________________________________________________________________

I think it might be crucial to leave the trail of research for each step directly there in the "modules" for the [CAA] Gallian text.   I do not wish to rewrite the text, but to EXPLORE, research and understand ... slowly, deeply, and even intensely.
« Last Edit: January 20, 2022, 03:31:59 pm by ... »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~