Fields whose Multiplicative Groups are Linear Spaces

Yuki Nakata

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arXiv · 1905.05714

Fields whose Multiplicative Groups are Linear Spaces

Yuki Nakata

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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 $\mathbb{Q}$.

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Yuki Nakata. 2019-05-09. Fields whose Multiplicative Groups are Linear Spaces. https://arxiv.org/abs/1905.05714

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