Bicomplex Algebraic Numbers

TL;DR

The paper studies algebraicity in the bicomplex algebra $\mathbb{B}$ and finite bicomplex extensions of $\mathbb{Q}$. It proves that every finite bicomplex extension $L$ is generated by a single $\omega$, so $L=\mathbb{Q}[\omega]$, and classifies when $L$ is a number field versus a direct sum $K_1\mathbf{e}_1+K_2\mathbf{e}_2$. It develops the arithmetic of these extensions by showing $\mathcal{O}_L=\mathcal{O}_{K_1}\mathbf{e}_1+\mathcal{O}_{K_2}\mathbf{e}_2$, with discriminant multiplicativity, a product formula for the zeta function $\zeta_L(s)=\zeta_{K_1}(s)\zeta_{K_2}(s)$, and a description of prime and irreducible elements. The paper provides explicit quadratic and quartic bicomplex extensions, including hyperbolic and Gaussian-type components, and demonstrates primes become semiprime in bicomplex rings, illustrating the applicability of the bicomplex framework.

Abstract

We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct classes of extensions depending on the nature of the generating element. We further establish a key decomposition property for bicomplex extensions, which serves as a foundation for studying their rings of integers. We also observe that prime elements in the ring of integers of a number field may become semiprime in the rings of integers of suitable bicomplex extensions. Finally, we present two explicit examples of finite bicomplex extensions.

Theorems & Definitions (35)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.1
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof