A Finer Classification of the Unit Sum Number of the Ring of Integers of Quadratic Fields and Complex Cubic Fields

Proceedings Mathematical Sciences

Nahid Ashrafi¹ 

[1] Department of Mathematics, Semnan University, Semnan, Iran$$

关键词: Unit;    unit sum number;    ring of integers;    quadratic fields;    complex cubic field.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

The unit sum number, $u(R)$, of a ring 𝑅 is the least 𝑘 such that every element is the sum of 𝑘 units; if there is no such 𝑘 then $u(R)$ is 𝜔 or ∞ depending on whether the units generate 𝑅 additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is 𝜔 or ∞. Then we present some examples showing that all possibilities can occur.

【 授权许可】

Unknown