Abstract
Let $\mathbb{P}^1(\overline{\mathbb{Q}})$ be the projective line over $\overline{\mathbb{Q}}$ and $H$ the Weil height on $\mathbb{P}^1(\overline{\mathbb{Q}})$. A classical result in algebraic number theory, so called Kronecker's theorem, states that $H(1,x)=1$ if and only if $x\in\overline{\mathbb{Q}}$ is 0 or a root of unity. In [4], Talamanca introduced some height functions on $M_n(\overline{\mathbb{Q}})$. The purpose of this paper is to show analogues of Kronecker's theorem for these heights: We determine height one matrices relative to these heights.
Acknowledgment
I deeply thank my master's advisor Yuichiro Takeda for his continued support.
I am really grateful to Valerio Talamanca for sending me a personal lecture note. It was very helpful when I wrote this paper.
Yuya Miyata told me some TeX commands which I used in the source code of this paper. I should thank him.
Finally, I am sincerely grateful to the anonymous referee for reading the draft carefully and giving me so many valuable comments. These did improve my ill-organized draft.
Citation
Masao Okazaki. "Height one matrices." Nihonkai Math. J. 30 (1) 19 - 26, 2019.
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