{"id":{"repo_id":"siu-theses","oai_identifier":"oai:opensiuc.lib.siu.edu:dissertations-1736"},"canonical_url":"https://search.dev.ndltd.org/etd/siu-theses/oai:opensiuc.lib.siu.edu:dissertations-1736","repository":{"repo_id":"siu-theses","name":"Southern Illinois University","base_url":"https://opensiuc.lib.siu.edu/do/oai/"},"display":{"title":"Explicit Factorization of Generalized Cyclotomic Polynomials of Order $2^m 3$ Over a Finite Field $F_q$","abstract":"We give explicit factorizations of $a$-cyclotomic polynomials of order $2^m 3$, $Q_{2^m3,a}(x)$, over a finite field $F_q$ with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of $F_q$. We use the relation between usual cyclotomic polynomials and $a$-cyclotomic polynomials. Factorizations split into eight categories according to $q \\equiv \\pm1$ (mod 4), $a$ and $-3$ are square in $F_q$. We find that the coefficients of irreducible factors are primitive roots of unity and in some cases that are related with Dickson polynomials.","abstract_html":"We give explicit factorizations of $a$-cyclotomic polynomials of order <span class=\"etd-inline-math\">2<sup>m</sup> 3</span>, <span class=\"etd-inline-math\">Q<sub>2<sup>m</sup>3,a</sub>(x)</span>, over a finite field <span class=\"etd-inline-math\">F<sub>q</sub></span> with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of <span class=\"etd-inline-math\">F<sub>q</sub></span>. We use the relation between usual cyclotomic polynomials and $a$-cyclotomic polynomials. Factorizations split into eight categories according to $q \\equiv \\pm1$ (mod 4), $a$ and $-3$ are square in <span class=\"etd-inline-math\">F<sub>q</sub></span>. We find that the coefficients of irreducible factors are primitive roots of unity and in some cases that are related with Dickson polynomials.","abstract_has_math":true,"creators":["Tosun, Cemile"],"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":"2013-08-01T07:00:00Z","date_published":"2013-08-01T07:00:00Z","updated_at":"2026-07-24T04:34:11Z","subjects":["Cyclotomic Polynomials","Dickson Polynomials","Factorization","Finite Field"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://opensiuc.lib.siu.edu/dissertations/733","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":["Tosun, Cemile"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-12-10T08:00:00Z"]},{"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":["Cyclotomic Polynomials","Dickson Polynomials","Factorization","Finite Field"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://opensiuc.lib.siu.edu/dissertations/733"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We give explicit factorizations of $a$-cyclotomic polynomials of order $2^m 3$, $Q_{2^m3,a}(x)$, over a finite field $F_q$ with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of $F_q$. We use the relation between usual cyclotomic polynomials and $a$-cyclotomic polynomials. Factorizations split into eight categories according to $q \\equiv \\pm1$ (mod 4), $a$ and $-3$ are square in $F_q$. We find that the coefficients of irreducible factors are primitive roots of unity and in some cases that are related with Dickson polynomials."]},{"key":"dc:title","label":"Title","values":["Explicit Factorization of Generalized Cyclotomic Polynomials of Order $2^m 3$ Over a Finite Field $F_q$"]}]}],"canonical_facts":{"dc:contributor":["Fitzgerald, Robert"],"dc:creator":["Tosun, Cemile"],"dc:date.available":["2015-12-10T08:00:00Z"],"dc:description.abstract":["We give explicit factorizations of $a$-cyclotomic polynomials of order $2^m 3$, $Q_{2^m3,a}(x)$, over a finite field $F_q$ with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of $F_q$. We use the relation between usual cyclotomic polynomials and $a$-cyclotomic polynomials. Factorizations split into eight categories according to $q \\equiv \\pm1$ (mod 4), $a$ and $-3$ are square in $F_q$. We find that the coefficients of irreducible factors are primitive roots of unity and in some cases that are related with Dickson polynomials."],"dc:identifier":["https://opensiuc.lib.siu.edu/dissertations/733"],"dc:subject":["Cyclotomic Polynomials","Dickson Polynomials","Factorization","Finite Field"],"dc:title":["Explicit Factorization of Generalized Cyclotomic Polynomials of Order $2^m 3$ Over a Finite Field $F_q$"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Only Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T04:34:11Z"}