New project time whilst I’m on winter break! This time I’m investigating cubic and biquadratic reciprocity, and already, as I’m about 2 weeks in, there’s so much modern algebra and stuff about finite fields that I have to either relearn or refresh my memory on. So far the most important part is understanding finite fields, both in terms of construction and existence. I’m nearly done learning about the basics of finite fields, so I then have to move on to understanding multiplicative characters (I have worked with quadratic multiplicative characters in my quadratic reciprocity project, but higher reciprocity demands cubic, biquadratic, \(n\)th degree multiplicative characters), as well as the stuff that comes with it such as Gauss sums. Then I can finally approach cubic and biquadratic reciprocity.

Maybe I’ll write a short expository paper about the journey to cubic and biquadratic reciprocity; we’ll see if I have time (If I do, I estimate that it will be perhaps 15-25 pages? I’m not quite sure yet, but it will be one of my larger projects, as it also hinges upon parts of my quadratic reciprocity project). This is kind of my last expository venture as from now on pretty much everything is going to be different, and original research will finally (kind of) be in my grasp!

Anyway, I will update this page when I either produce or don’t produce something. I am keeping detailed notes with insights and proofs from other areas (cough cough mostly algebra), so if I don’t end up writing an expository paper I suppose I can just provide those.

(Updated 24 December, 2023)