CONNECTIONS AND IMPLICATIONS AMONG THE GELFAND-MAZUR, FUNDAMENTAL THEOREM OF ALGEBRA AND LIOUVILLE'S THEOREM

Abstract

The study of fundamental theorems in functional analysis often highlights deep interconnections between results that, at first glance, may appear independent. In this paper, we investigate the relationships between three classical results: Gelfand-Mazur Theorem, Liouville's Theorem and Fundamental Theorem of Algebra. While previous 
works have explored equivalences between pairs of these theorems, a systematic study of their mutual implications remains less developed. This paper aims to systematically explore the equivalences between these three theorems, as the literature often compares only two of them at a time.  Here, we seek to highlight their mutual connections simultaneously, providing a broader understanding of their interplay. Our approach builds upon established results and techniques, offering a unified perspective that sheds light on their deeper structure.

Keywords: Gelfand-Mazur, Liouville, Fundamental Theorem of Algebra, Complex Analysis.