Please use this identifier to cite or link to this item: https://idr.l4.nitk.ac.in/jspui/handle/123456789/17757
Title: Galois Group of Certain Algebraic Extensions and Their Relations With Primes In Arithmetic Progression
Authors: Sahu, Sehra
Supervisors: Shankar, B R
Keywords: Galois Group;Multi-Quadratic Extension;Cyclotomic Extension;Residue Pattern
Issue Date: 2023
Publisher: National Institute Of Technology Karnataka Surathkal
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.
URI: http://idr.nitk.ac.in/jspui/handle/123456789/17757
Appears in Collections:1. Ph.D Theses

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