12

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they specialize in mathematics at the college level, they do not learn any more. Why not? Because we have computers now, so most people do not need to solve polynomial ...


9

(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than one data point) this was a standard topic that showed up somewhere in the standard first year (2-semester) graduate algebra sequence (e.g. Lang, Hungerford, ...


9

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the ...


8

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational ...


7

Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right? When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year. The introduction to abstract algebra used "Numbers, Groups, and Codes", ...


6

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples: Lagrange's theorem in abstract algebra: The order of a subgroup $H$ of a finite group $G$ divides the order of $...


4

For all $n \geq 3$ the alternating group $A_{n}$ can be realized as the orientation-preserving symmetries of the regular $n$-simplex (triangle, tetrahedron, etc.). In this case the full symmetry group is $S_{n}$, and the relation between $A_{n}$ and $S_{n}$ can be seen geometrically. For instance, when $n = 3$, $A_{3}$ comprises the rotations through an ...


4

Well, certainly nearly all the usual texts (here is an open source one!) do this via necklace/bracelet examples, and you can definitely get the classes this way. The treatment in Anderson's Discrete Math would be typical - coloring an $n$-sided polygon with $m$ colors. Even with two colors and four sides becomes nontrivial and leads you to stabilizers etc. ...


4

Permit me to amplify @XanderHenderson's comment pointing to Nathan Carter's Visualizing Group Theory, which I think is the perfect vehicle for your intentions. The MESE question, What makes cosets hard to understand? pointed to his illustration of cosets in $A_4$ over several slides, of which this is one (Download .ppt):       ...


4

Since you are doing this geometrically, I would suggest a different tact on your treatment of normal subgroups: You can think of them as stabilizers of certain extra decorations. For example, in a dihedral group we could label the various configurations with a right or left handed orientation. Then you obtain the (normal) rotation subgroup as the stabilizer ...


4

Synthetic division is a standard part of the stereotypical "algebra 2" course in the US (~grade 11) and is normally covered including drill problems and examination. In my experience, cubics and quartics are in the algebra 2 book (I just checked and they are in mine) but in sections with an asterisk. Typically the asterisk sections are not covered because ...


4

Most math majors never learn cardano's methods. As for synthetic division, it is part of most high school curricula and linear algebra courses. Cubic and quartic equations aren't part of any curriculum I'm aware of, not because of computers, but because students should learn "more important topics" and because the solutions of these equations are very long ...


3

Search [mathematics.se] using the (reference-request) tag, and the (abstract-algebra) tag. There have been a number of posts, from students at various levels of study in abstract algebra, seeking text recommendations. One such post, mentions Lang's Algebra. It's a very thorough text (scan the Table of Contents here, but doesn't cover category theory ...


3

http://www.csun.edu/~asethura/GIAAFILES/GIAAV1.0/GIAAV1.0.pdf "A Gentle Introduction to Abstract Algebra" this text is pretty good (I used it in my undergraduate algebra course), starts with rings, and uses a distribution license similar to that of linux


2

I could to an entirely abstract example, with actual squares or triangles, with painting or marking the corners or sides, but I don't think that would be suitable as a motivational example. I think this could be a motivational example. True Story: My daughter and I tried to make a little card game once (when we were tired of the ones we owned), and ...


2

Permit me to endorse @halfbloodprince's recommendation.           Stillwell's four pillars are: Euclidian straight-edge/compass constructions, linear algebra & coordinates, projective geometry, transformational groups & non-Euclidean geometry. The most important aspect is that he views geometry from several different ...


2

I think you are looking for a "French style" approach. So maybe Geometry by M. Audin is good for you. Note that a new edition in French is available. It covers pretty well points 2 and 3 but not 1. Also interesting is the book of P. Gabriel, Matrizen,Geometrie, Lineare Algebra, unfortunatedly only available in German or a French translation. I hope this ...


2

In a math department (not physics department) I'd introduce it after some more familiar constructions (quotient groups, cyclic groups with a choice of generator,...) have already been described a second time by a universal mapping property. Tensor products will be tough the first time no matter what, so it would help if the idea of a construction based on a ...


2

Interesting question, here are some ideas just off the top of my head: Direct product groups act on Cartesian products, so the symmetry group of two two squares rotating independently (you could then also talk about how this is a subgroup of the full symmetry group where you're also allowed to swap the squares if they're identical.) Cosets: This isn't great,...


1

I taught myself the basic material from the first 6 chapters of the 2nd edition of Rotman's Advanced Modern Algebra, and I loved it and felt things were explained clearly. Earlier I had tried teaching myself out of Artin's Algebra but got stuck or spent too much time trying to understand with details of the wallpaper group (which in retrospect I wish I hadn'...


1

There are some references to using the geometric art of M.C. Escher get across group theory (and other) ideas. For example this article of Marjorie Senechal: https://upcommons.upc.edu/bitstream/handle/2099/1134/st15-07-a3-ocr.pdf and https://files.eric.ed.gov/fulltext/ED583835.pdf More generally you can look for papers related to using the wallpaper ...


1

The 15 puzzle is a nice example of an alternating group. The classic puzzle gives a concrete model for $A_{15}$. Generalizations extend it to $A_{n-1}$ for any composite $n$. Students could work out the way in which it is a group, and then parity could be introduced as an answer to the question of which configurations of the puzzle board are reachable from ...


1

This is an old question now, but I would toot a horn and suggest that perhaps basic finite group representation theory would fit in well, given that there are applications to noncooperative and cooperative game theory as well as social choice. There are many wonderful areas of algebra that have good connections to things LSE cares about.


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