Muhammad, Aslam Malik (2002) GROUP ACTIONS ON FIELDS. Doctoral thesis, University of the Punjab.

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Coset Diagrams are a graphical representation of the permutation action of the groups. Studying groups through their actions on different sets and algebraic structure has become a useful technique to know about the structure of the groups. The graphs have played a vital role in studying these actions. The main object of this work is to examine the actions of infinite groups G = < x,y; x2 = y3 =1>, H = < x, y; x2 = y4 = 1> and M = < x, y; x2 = y6 = 1> on real quadratic fields Q(√n) and to find the subsets of Q(√n) invariant under the action of each of these groups. Certain proper subsets of Q(√n) invariant under the actions of each of these groups G, H and M, Have also been discussed in this dissertation. In this dissertation, a type of graphs, called coset diagrams, is employed to investigate the orbits of certain subsets Q*(√p), Q//(√p) and Q///(√p) of Q(√p), p a rational prime, under the actions of the groups G, H and M respectively. This dissertation is concerned with the determination of number of ambiguous numbers, ambiguous integers, ambiguous units and ambiguous primes in certain subsets Q*(√n), Q//(√n) and Q///(√n) of Q(√n) which are invariant under the action of the groups G, H an M respectively. One of the principal results of this dissertation is that we have determined, for each non square positive rational integer n, the actual number of ambiguous numbers in Q*/(√n), as a function of n.

Item Type: Thesis (Doctoral)
Uncontrolled Keywords: coset diagrams, ambiguous numbers, ambiguous integers, ambiguous units, ambiguous primes, graphical representation, quadratic field
Subjects: Q Science > QA Mathematics
Depositing User: Muhammad Khan Khan
Date Deposited: 25 Oct 2016 07:03
Last Modified: 25 Oct 2016 07:03

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