{"id":15102,"date":"2022-06-18T19:14:16","date_gmt":"2022-06-18T23:14:16","guid":{"rendered":"https:\/\/carleton.ca\/scs\/?page_id=15102"},"modified":"2026-06-08T14:57:31","modified_gmt":"2026-06-08T18:57:31","slug":"tr-228-power-roots-of-polynomials-over-arbitrary-fields","status":"publish","type":"page","link":"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1993\/tr-228-power-roots-of-polynomials-over-arbitrary-fields\/","title":{"rendered":"TR-228: Power Roots of Polynomials over Arbitrary Fields"},"content":{"rendered":"\n<section class=\"w-screen px-6 cu-section cu-section--white ml-offset-center md:px-8 lg:px-14\">\n    <div class=\"space-y-6 cu-max-w-child-5xl  md:space-y-10 cu-prose-first-last\">\n\n            <div class=\"cu-textmedia flex flex-col lg:flex-row mx-auto gap-6 md:gap-10 my-6 md:my-12 first:mt-0 max-w-5xl\">\n        <div class=\"justify-start cu-textmedia-content cu-prose-first-last\" style=\"flex: 0 0 100%;\">\n            <header class=\"font-light prose-xl cu-pageheader md:prose-2xl cu-component-updated cu-prose-first-last\">\n                                    <h1 class=\"cu-prose-first-last font-semibold !mt-2 mb-4 md:mb-6 relative after:absolute after:h-px after:bottom-0 after:bg-cu-red after:left-px text-3xl md:text-4xl lg:text-5xl lg:leading-[3.5rem] pb-5 after:w-10 text-cu-black-700 not-prose\">\n                        TR-228: Power Roots of Polynomials over Arbitrary Fields\n                    <\/h1>\n                \n                                \n                            <\/header>\n\n                    <\/div>\n\n            <\/div>\n\n    <\/div>\n<\/section>\n\n\n\n<p>Carleton University<br><a href=\"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1993\/\">Technical Report<\/a>&nbsp;<strong>TR-228<\/strong><br>September 1993<\/p>\n\n\n\n<h2 id=\"power-roots-of-polynomials-over-arbitrary-fields\" class=\"wp-block-heading\">Power Roots of Polynomials over Arbitrary Fields<\/h2>\n\n\n\n<p>Vincenzo Acciaro<\/p>\n\n\n\n<h3 id=\"abstract\" class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p>Let F be an arbitrary field, and f(x) a polynomial in one variable over F of degree 2: 1. Given a polynomial g( x) #- 0 over F and an integer m &gt; 1 we give necessary and sufficient conditions for the existence of a polynomial z(x) E F[x] such that z(xr = g(x) (mod f(x)). We show how our results can be specialized to IR, 4J and to finite fields. Since our proofs are constructive it is possible to translate them into an effective algorithm when F is a computable field (e.g. a finite field or an algebraic number field).<\/p>\n\n\n\n<p><a href=\"https:\/\/carleton.ca\/scs\/wp-content\/uploads\/sites\/260\/TR-228.pdf\">TR-228.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Carleton UniversityTechnical Report&nbsp;TR-228September 1993 Power Roots of Polynomials over Arbitrary Fields Vincenzo Acciaro Abstract Let F be an arbitrary field, and f(x) a polynomial in one variable over F of degree 2: 1. Given a polynomial g( x) #- 0 over F and an integer m &gt; 1 we give necessary and sufficient conditions for [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":11912,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_cu_dining_location_slug":"","footnotes":"","_links_to":"","_links_to_target":""},"cu_page_type":[],"class_list":["post-15102","page","type-page","status-publish","hentry"],"acf":{"cu_post_thumbnail":""},"_links":{"self":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/comments?post=15102"}],"version-history":[{"count":2,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15102\/revisions"}],"predecessor-version":[{"id":24445,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15102\/revisions\/24445"}],"up":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/11912"}],"wp:attachment":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/media?parent=15102"}],"wp:term":[{"taxonomy":"cu_page_type","embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/cu_page_type?post=15102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}