Open Access
2019 Height one matrices
Masao Okazaki
Nihonkai Math. J. 30(1): 19-26 (2019).


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.


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.


Download Citation

Masao Okazaki. "Height one matrices." Nihonkai Math. J. 30 (1) 19 - 26, 2019.


Received: 4 March 2019; Revised: 1 June 2019; Published: 2019
First available in Project Euclid: 17 October 2019

zbMATH: 07155345
MathSciNet: MR4019890

Primary: 11G50
Secondary: 11C20 , 15A60 , 16S50

Keywords: height one matrices , heights of square matrices

Rights: Copyright © 2019 Niigata University, Department of Mathematics

Vol.30 • No. 1 • 2019
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