Galois Group of Certain Algebraic Extensions and Their Relations With Primes In Arithmetic Progression

Abstract

Explicit structure of Galois group of Q( a1 , a2 , ..., an ) over Q was calculated by Karthick Babu and Anirban Mukhopadhyay. Expanding this knowledge, the problem of finding an ex- √ √ √ plicit Galois group of the field extension Q( a1 , a2 , ..., an , ζd ) over Q in terms of its action √ on ζd and ai for 1 ≤ i ≤ n has been studied. Let p be an odd prime. If we have an integer g which generates a subgroup of index t in (Z/pZ)∗ , then we call g to be a t-near primitive root modulo p. Pieter Moree and Min Sha showed that each coprime residue class contains a positive density of primes p not having g as a t-near primitive root. In this note, for a subset {a1 , a2 , . . . , an } of Z \ {0}, we shall prove that each such coprime residue class contains a positive density of primes p such that ai is not a t-near primitive root. Additionally, ai ’s satisfy certain residue pattern modulo p, for 1 ≤ i ≤ n.