8

Not sure if this counts for the OP's purpose, but I started with Hungerford's Abstract Algebra: An Introduction. The first 3 chapters go like this: Arithmetic in $\mathbb Z$ revisited Congruence in $\mathbb Z$ and modular arithmetic (i.e., $\mathbb Z_n$) Rings Etc. The motivating thesis is explicitly laid out in the Preface and a Thematic Table of ...


6

Typically, students learn about finite groups long before they think of "variety of algebras" or "compactness theorem". I suppose you also want to exclude fields from "algebra"? I would say: topics for an algebra syllabus are chosen according to how useful they are; not according to how they fit into some general framework.


6

I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I think there is lots of variation. Typically groups are in the first course, but Sylow theorems might be in the second course, along with rings. And the second ...


5

How about this: Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$. Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of degree $n \geq 1$. Choose distinct elements $z_1, z_2, z_3, \dots, z_{n+1}$ of $k$. By repeated use of the linear factor theorem, we know that $f(x) = (x-z_1)(x-z_2)(...


4

A nice example is Paolo Aluffi's Algebra: Chapter 0. The first chapter (40 pages) is Preliminaries: Set theory and categories. Groups and rings follow in the next three chapters. This book is special as it follows a cathegorically minded viewpoint (in the author's words), which I like very much. According to the introduction, the book is suitable for upper-...


3

I think this certainly makes a lot more sense than teaching rings before groups. I think it might be a wash whether you teach all of the parts of groups and then all of the parts of rings and note the similarities along the way or if you collate them like this book does, but at least this seems pedagogically viable. I think my larger concern is how the ...


3

The Jordan Curve Theorem is the canonical example of this. Also, the fact that the sum of the first n odd numbers is n squared is obvious from the well-known ‘wrapping’ diagram, but the actual proof is via mathematical induction, which is a challenge for many students. Also: The theorem on sphere-packing, long time in coming, merely confirmed what what ...


2

This is a little long for a comment so it will have to be an answer. First, you should talk to your professor; I'm pretty sure that they will be able to definitively clear your issue up with a small amount of back-and-forth discussion. You seem to have trouble starting around your encounter with the word "define". I'm not really sure what we are ...


1

Re (1), you should lower your standards. As previously discussed, the population for Math55 in high school is tiny. (Add onto that the difficulty of finding them and it just makes little sense to try to develop this.) In addition, you really lack the math knowledge OR the practical pedagogical experience to develop Math55 for high school (even the time, ...


1

For (2), I'd suggest creating the lessons first as blog posts or youtube videos and then sharing them. I think the math reddit might be a good place, but I'm not active there. These can then in turn serve as advertisements for any active curricula you'd like to implement. For (3): Start with an overall structure of the course. Make a list of course-level ...


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