{"id":{"repo_id":"siu-theses","oai_identifier":"oai:opensiuc.lib.siu.edu:dissertations-1773"},"canonical_url":"https://search.dev.ndltd.org/etd/siu-theses/oai:opensiuc.lib.siu.edu:dissertations-1773","repository":{"repo_id":"siu-theses","name":"Southern Illinois University","base_url":"https://opensiuc.lib.siu.edu/do/oai/"},"display":{"title":"Higher-Order Lucas Sequences and Dickson Polynomials","abstract":"In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.","abstract_html":"In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.","abstract_has_math":false,"creators":["Boone, Joshua Daniel"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Campus Only Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Fitzgerald, Robert"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-12-01T08:00:00Z","date_published":"2013-12-01T08:00:00Z","updated_at":"2026-07-24T04:34:20Z","subjects":["Algebra","Dickson","Lucas","Polynomials","Sequences"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://opensiuc.lib.siu.edu/dissertations/770","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Fitzgerald, Robert"]},{"key":"dc:creator","label":"Author","values":["Boone, Joshua Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Only Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Algebra","Dickson","Lucas","Polynomials","Sequences"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://opensiuc.lib.siu.edu/dissertations/770"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5."]},{"key":"dc:title","label":"Title","values":["Higher-Order Lucas Sequences and Dickson Polynomials"]}]}],"canonical_facts":{"dc:contributor":["Fitzgerald, Robert"],"dc:creator":["Boone, Joshua Daniel"],"dc:description.abstract":["In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5."],"dc:identifier":["https://opensiuc.lib.siu.edu/dissertations/770"],"dc:subject":["Algebra","Dickson","Lucas","Polynomials","Sequences"],"dc:title":["Higher-Order Lucas Sequences and Dickson Polynomials"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Only Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T04:34:20Z"}