Density of Primes in 𝑙-th Power Residues

Proceedings Mathematical Sciences

Prem Prakash Pandey¹  R Balasubramanian² 

[1] $$;The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 00 , India$$

关键词: Distribution of primes;    higher residue symbols.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Given a prime number 𝑙, a finite set of integers $S={a_1,ldots,a_m}$ and 𝑚 many 𝑙-th roots of unity $𝜁^{r_i}_l,i=1,ldots,m$ we study the distribution of primes 𝑝 in $mathbb{Q}(𝜁_l)$ such that the 𝑙-th residue symbol of $a_i$ with respect to 𝑝 is $𝜁^{r_i}_l$, for all 𝑖. We find out that this is related to the degree of the extension $mathbb{Q}left(a^{frac{1}{l}}_1,ldots,a^{frac{1}{l}}_might)/mathbb{Q}$. We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from $S={a_1,ldots,a_m}$. This latter argument enables one to describe the degree $mathbb{Q}left(a^{frac{1}{l}}_1,ldots,a^{frac{1}{l}}_might)/mathbb{Q}$ in much simpler terms.

【 授权许可】

Unknown