On the complex Levi-Civita field: algebraic and topological structures, and foundations for analysis

Document Type : Original Article

Author

Khodr Shamseddine

Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

10.22034/kjm.2023.378706.2735

Abstract

In this paper, we will introduce the complex Levi-Civita field $\mathcal{C}$. We start by reviewing the algebraic structure of the field; in particular, $\mathcal{C}$ is the smallest non-Archimedean valued field extension of the complex numbers field $\mathbb{C}$ that is algebraically closed and complete in the valuation topology.
Two topologies on $\mathcal{C}$ will be studied in detail: the valuation topology induced by a non-Archimedean valuation on the field, and another weaker topology induced by a family of seminorms, which we will call weak topology. We show that each of the two topologies results from a metric on $\mathcal{C}$ and that the valuation topology is not a vector topology while the weak topology is. Then we give simple characterizations of open, closed, and compact sets in both topologies.
Finally, we define continuity and differentiability for a $\mathcal{C}$-valued function at a point or on a subset of $\mathcal{C}$, we present key results for such functions, and we set the foundations for a Cauchy-like analysis theory on the field $\mathcal{C}$.

Keywords

Complex Levi-Civita Field

valuation topology

weak topology

linear topological space

non-Archimedean complex analysis

Subjects

• 46 Functional analysis

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