{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-1901"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-1901","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"Primes of the form X² + nY² in function fields","abstract":"Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the appropriate notion of Hilbert class field in this context. In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S.Dummit and D.Hayes for the case where -n is monic of odd degree.","abstract_html":"Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the appropriate notion of Hilbert class field in this context. In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S.Dummit and D.Hayes for the case where -n is monic of odd degree.","abstract_has_math":false,"creators":["Maciak, Piotr"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-01-01T08:00:00Z","date_published":"2010-01-01T08:00:00Z","updated_at":"2026-07-24T02:58:40Z","subjects":["weak representation","strong representation","principal form","normalizing field","j-invariant"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-04272010-174411","https://repository.lsu.edu/gradschool_dissertations/902"],"render_values":[{"text":"etd-04272010-174411","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/902","href":"https://repository.lsu.edu/gradschool_dissertations/902","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.902","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Maciak, Piotr"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010-04-16"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:10:22Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["weak representation","strong representation","principal form","normalizing field","j-invariant"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-04272010-174411","10.31390/gradschool_dissertations.902","https://repository.lsu.edu/gradschool_dissertations/902"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the appropriate notion of Hilbert class field in this context. In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S.Dummit and D.Hayes for the case where -n is monic of odd degree."]},{"key":"dc:title","label":"Title","values":["Primes of the form X² + nY² in function fields"]}]}],"canonical_facts":{"dc:creator":["Maciak, Piotr"],"dc:date":["2010-04-16"],"dc:date.available":["2022-05-12T23:10:22Z"],"dc:description.abstract":["Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the appropriate notion of Hilbert class field in this context. In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S.Dummit and D.Hayes for the case where -n is monic of odd degree."],"dc:identifier":["etd-04272010-174411","10.31390/gradschool_dissertations.902","https://repository.lsu.edu/gradschool_dissertations/902"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["weak representation","strong representation","principal form","normalizing field","j-invariant"],"dc:title":["Primes of the form X² + nY² in function fields"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:58:40Z"}