UPDATE (7 July, 2024): The paper is now on this site as well as on the arXiv. For the version on this website, check here. For the version on the arXiv, check here.

Hello! It has been a while since I have last posted, but here I am now! A few months ago (January or so) I started working on a paper related to the study of Galois fields (finite fields) and higher reciprocity. Since then I have written nearly 45 pages, and it is quite nearly complete. Over these few months, I have furthered my mathematical maturity, taken advice and learned a lot from both peers and professors, and experienced a few revelations here and there in relation to some topics or proofs. I also created a poster and gave a sort of informal presentation on the paper! Writing is hard, and when you’re busy it’s difficult to make significant progress. I think I initially said that I would complete this by the end of January; then the end of March; then May; and now look, it’s already June. I wouldn’t say this is procrastination at all - rather a combination of business and a quest for greater mathematical maturity.

Even while this paper is not fully complete, I will provide the abstract for any who are interested in reading about what is covered in the paper.

Abstract

Cubic and biquadratic reciprocity have long since been referred to as ``the forgotten reciprocity laws”, largely since they provide special conditions that are widely considered to be unnecessary in the study of number theory. However, this paper aims to approach reciprocity with ample detail to motivate its existence. In this exposition of finite fields and higher reciprocity, we will begin by introducing concepts in abstract algebra and elementary number theory. This will motivate our approach toward understanding the structure and then existence of finite fields, especially with a focus on understanding the multiplicative group \(\mathbb{F}^{*}\). While surveying finite fields we will provide another proof of quadratic reciprocity. We will proceed to investigate properties of the general multiplicative character, covering the concept of a general Gauss sum as well as basic notions of the Jacobi sum. From there we will begin laying the foundations for the cubic reciprocity law, beginning with a classification of the primes and units of the Eisenstein integers, denoted \(\mathbb{Z}[\omega]\), and further investigations into the residue class ring \(\mathbb{Z}[\omega]/\pi\mathbb{Z}[\omega]\) for \(\pi\) prime, which is predominantly the world in which cubic reciprocity lies.

We will then use multiplicative characters to define the cubic residue character and state cubic reciprocity in its entirety. Following this, we provide a proof of the cubic reciprocity law as well as its supplementary theorems using cubic Gauss sums. We will finish the section on cubic reciprocity with a brief survey of the cubic residue character of the even prime \(2\) and state a significant result due to Gauss that summarizes the conditions for \(2\) to be a cubic residue.

We conclude with the statement of biquadratic reciprocity and provide a brief discussion on how it relates to cubic reciprocity in both its proof and usage of the analogy between the Eisenstein integers, \(\mathbb{Z}[\omega]\), and the Gaussian integers, \(\mathbb{Z}[i]\).

When the completed paper appears, I will reference it here. I will likely put it on the arXiv. (it is now there at the top of this page)

UPDATE (25 June, 2024): I have decided that I will submit a section of this paper to either the Mathematics Magazine or the College Mathematics Journal . To prepare for submission I will need to edit one particular section to be in a more expository form with plenty of examples, and I might even need to shorten it since it is currently 15 pages, and the alleged limit for either magazine is approximately 14 pages. (25 June, 2024)

(Updated 7 July, 2024)