• Prem Prakash Pandey

      Articles written in Proceedings – Mathematical Sciences

    • Density of Primes in 𝑙-th Power Residues

      R Balasubramanian Prem Prakash Pandey

      More Details Abstract Fulltext PDF

      Given a prime number 𝑙, a finite set of integers $S=\{a_1,\ldots,a_m\}$ and 𝑚 many 𝑙-th roots of unity $\zeta^{r_i}_l,i=1,\ldots,m$ we study the distribution of primes 𝑝 in $\mathbb{Q}(\zeta_l)$ such that the 𝑙-th residue symbol of $a_i$ with respect to 𝑝 is $\zeta^{r_i}_l$, for all 𝑖. We find out that this is related to the degree of the extension $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$. 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 $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$ in much simpler terms.

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