The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur’s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.
Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results.
","abstract_html":"<p>The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur’s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.</p><p>Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results.</p>","abstract_has_math":false,"creators":["Mcwhorter, Caleb"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Steven Diaz"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-05-22T07:00:00Z","date_published":"2021-05-22T07:00:00Z","updated_at":"2026-07-24T04:56:04Z","subjects":["elliptic curves","Galois field","nonic field","odd degree","torsion subgroups","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/1322","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Steven Diaz"]},{"key":"dc:creator","label":"Author","values":["Mcwhorter, Caleb"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["elliptic curves","Galois field","nonic field","odd degree","torsion subgroups","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/1322"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur’s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.
Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results.
"]},{"key":"dc:title","label":"Title","values":["Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields"]}]}],"canonical_facts":{"dc:contributor":["Steven Diaz"],"dc:creator":["Mcwhorter, Caleb"],"dc:description.abstract":["The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur’s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.
Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results.
"],"dc:identifier":["https://surface.syr.edu/etd/1322"],"dc:subject":["elliptic curves","Galois field","nonic field","odd degree","torsion subgroups","Physical Sciences and Mathematics"],"dc:title":["Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:56:04Z"}