Fields whose Multiplicative Groups are Linear Spaces
Number Theory
2021-10-19 v2
Abstract
The purpose of this paper is to study fields whose multiplicative groups admit the structure of linear spaces. We prove that the multiplicative group of a finite field is a linear space if and only if the order of the multiplicative group is 1, 2, or a Mersenne prime. We give necessary conditions for the multiplicative group of an infinite field to be a linear space over another field. We also construct an example of an infinite field whose multiplicative group is a linear space over .
Cite
@article{arxiv.1905.05714,
title = {Fields whose Multiplicative Groups are Linear Spaces},
author = {Yuki Nakata},
journal= {arXiv preprint arXiv:1905.05714},
year = {2021}
}
Comments
The content of this paper is based on the author's presentation at the 8th Science Intercollegiate Contest held on March 2-3, 2019, hosted by Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan