There's a couple of new/original results scattered throughout the paper, but mostly it ends up being a list of classical results.
My intention with this paper was originally to delve into some Algebraic Geometry-flavoured research for gL, but I ended up feeling like it was better to be more general and get some basic Algebraic facts that can provide a basis for applications of gL to wider fields, and some algebraic facts that would provide more depth and formalization to past research.
This one was pretty fun to write, and it was appropriate, difficulty-wise, for a summer that saw me in a panic to find housing after moving across the country for the 2nd time in 9 months (amongst other things).
Anyway, all things considered, I'm pretty pleased with the resulting paper. An errata will be shortly forthcoming, I imagine.
A note: I put Stony Brook down as my institution because, technically and at this moment, it's still my institutional affiliation. I added "various Oregon Wilderness Areas" because I largely wrote this paper during camping trips in Mt. Washington, Mt. Jefferson, and Three Sisters Wilderness areas in Oregon.
Update (12 Sept, 2022)
A couple of non-errata things to add:
1) I want to give a special acknowledgement to (in alphabetical order) Prof. Kamenova, Prof. Laza, Prof McClean, and Prof Viro, who taught the courses I took last academic year at SBU. The info I gleaned from those courses was invaluable, both for writing this paper, and in general
I'd especially like to acknowledge the fact that I probably would have gone the rest of my life thinking the Circle Group was a trivial group without Prof. McClean mentioning that the Circle Group was in fact isomorphic with the reals modulo the integers during a class. I had previously been given some bad info, and I'm glad it was corrected.
As always, any misinterpretation of facts is solely attributable to me, and not to any Profs.
2) The section on rings is especially short because I was (as mentioned above) initially trying to write an Algebraic Geometry-flavoured paper. I got a little turned around trying to show a few things, had to start over, and the section on rings suffered the most.
3)I omitted a lot of proofs of otherwise classical results. Writing out whole detailed proofs for things like "the circle group is isomorphic to the real modulo the integers" seemed like overkill (and would have resulted in a MUCH longer paper). My thought in omitting such proofs was that the results are so well known that I wouldn't be adding anything by including the proofs.
1) Writing the circle as "x^{2}+y^{2}-1=0" was meant to imply a conic derived from letting 1=1^{2}=z^{2}. I didn't explicitly state that, but I do feel like I'm novice enough where it's probably important to explicitly state that. Thank goodness for errata's and the ability to edit blog posts.