The solvability of the cubic congruence x^3≡2 (mod p) is referred to as the cubic character of 2. In evaluating the cubic character of 2, we introduce the Eisenstein integers, Gauss and Jacobi sums, and the law of cubic reciprocity. We motivate this proof by giving ample historical information surrounding the early development of higher reciprocity laws as well as the Gauss’ proof of the solvability of the quadratic congruence x^2≡2 (mod p); conventionally the quadratic character of 2. We simultaneously outline other relevant contributions by Fermat, Euler, Legendre, Jacobi, and Eisenstein.
@article{relmat2024quadcub2,title={The Quadratic and Cubic Characters of 2},author={Carl Relyea, Matias},year={2025},note={This paper has been submitted for re-review in the Mathematics Magazine (MAA).},}
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 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 Z[ω], and further investigations into the residue class ring Z[ω]/πZ[ω] for π 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, Z[ω], and the Gaussian integers, Z[i].
@article{relmat2024finiterec,title={On Finite Fields and Higher Reciprocity},author={Carl Relyea, Matias},year={2024},eprint={2407.03559},archiveprefix={arXiv},primaryclass={math.NT}}
In this expository paper, we will outline several interesting properties of the q-factorial and its relationship with inversions, the q-derivative and the q-analogue of Taylor’s theorem, a theorem about the uniqueness of q-antiderivatives up to a constant alongside evaluation of definite q-integrals, as well as several perspectives and proofs of the q-binomial theorem. We will conclude with an introduction to the q-Pochhammer symbol, and use it to define the q-hypergeometric function and prove an interesting fact about the special q-hypergeometric series 1Φ0.
As one of the most important theorems in Elementary Number Theory, the Law of Quadratic Reciprocity is both incredibly beautiful and rewarding to explore. In this paper, we will explore characteristics of quadratic residues and quadratic nonresidues, and apply them to varying conditions on primes. After that, we will illustrate and prove a statement of equivalence of the Law of Quadratic Reciprocity using an assertion by Euler. Following this, we offer three unique proofs of the Law of Quadratic Reciprocity using Eisenstein’s Lemma and lattice-point counting, primitive nth roots of unity, and Quadratic Gauss Sums. Finally, we give a brief introduction to a generalization of the Legendre Symbol known as the Jacobi Symbol, and then prove Fermat’s Theorem on the Sum of two Squares using the Law of Quadratic Reciprocity and the Method of Infinite Descent.
@article{relmat2022quadrec,title={Proofs and Applications of Quadratic Reciprocity},author={Carl Relyea, Matias},year={2022},publisher={Academia.edu},note={This was completed as a part of the Euler Circle.}}
In this rather brief paper, we will define and prove several interesting properties of the Quadratic Gauss Sum, and eventually end with the main theorem of this paper: the sign of the Quadratic Gauss Sum. The proof that we will provide is one by Kronecker, although there are many more that can be interesting. This is not a generalization to Gauss Sums, but we will utilize the conventional notation with the Dirichlet Character χ for the sake of functionality.
@article{relmat2022quadgausssum,title={On the Value of the Quadratic Gauss Sum},author={Carl Relyea, Matias},year={2022},publisher={Academia.edu},}
An Introduction to Algebraic Numbers and Algebraic Integers
In this brief introduction to algebraic numbers and algebraic integers, we will explore some properties of finite-dimensional vector spaces, and through the language of algebraic numbers and algebraic integers, state and prove several fundamental results in field theory and ring theory; namely that the set of algebraic numbers forms a field and that the set of algebraic integers forms a ring. These structures are related to and stem from their respective algebraic number fields, known as Q modules and Z modules respectively.
@article{relmat2022algnumbint,title={An Introduction to Algebraic Numbers and Algebraic Integers},author={Carl Relyea, Matias},year={2022},publisher={Academia.edu},}
In this paper we will define the Bézier curve and introduce several interesting properties. We will also introduce the Bernstein polynomial and Bernstein basis polynomials, and state how they relate to De Casteljau’s algorithm and rational Bézier curves. We will conclude with a short derivation of the derivative of the Bézier curve, alongside an outline for a method of calculating the derivative of a Bézier curve at a given point.
@article{relmat2022briefbezier,title={A Brief Introduction to Bézier Curves},author={Carl Relyea, Matias},year={2022},publisher={Academia.edu},}
-480.webp)
