# Fields whose Multiplicative Groups are Linear Spaces

## TL;DR

This paper investigates when the multiplicative groups of fields can be structured as linear spaces, establishing conditions for finite and infinite fields, including specific classifications and examples.

## Contribution

It characterizes finite fields with multiplicative groups as linear spaces and provides conditions and examples for infinite fields.

## Key findings

- Finite field multiplicative group is a linear space iff its order is 1, 2, or a Mersenne prime.
- Necessary conditions are given for infinite fields' multiplicative groups to be linear spaces.
- An example of an infinite field with a multiplicative group as a linear space over 4

## 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 $\mathbb{Q}$.