{"id":11777,"date":"2019-12-03T16:24:11","date_gmt":"2019-12-03T21:24:11","guid":{"rendered":"https:\/\/library.bc.edu\/answerwall\/?p=11777"},"modified":"2019-12-05T11:42:50","modified_gmt":"2019-12-05T16:42:50","slug":"cube-root-unity-in-x2x1","status":"publish","type":"post","link":"https:\/\/library.bc.edu\/answerwall\/2019\/12\/03\/cube-root-unity-in-x2x1\/","title":{"rendered":"cube root unity in x^2+x+1"},"content":{"rendered":"\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" width=\"320\" height=\"310\" src=\"https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/aw120419-5.jpg\" alt=\"symbols in this post--a set of conditions describing x^2 +x+1, cannot be displayed due to technical problems.\" class=\"wp-image-11762\" srcset=\"https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/aw120419-5.jpg 320w, https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/aw120419-5-300x291.jpg 300w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><figcaption>\u03c9 \u2208 C cube root unity, P prime. Then (P) \u2208 Z[\u03c9] maximal, \u21d4 in Fp has no solution to x^2+x+1?<\/figcaption><\/figure>\n\n\n\n<p>My advanced math assistants have not yet responded; I hope to have an answer tomorrow. In the meantime, I&#8217;d like to clarify whether I&#8217;ve represented your post accurately in typeset: \u03c9 \u2208 C cube root unity, P prime. Then (P) \u2208 Z[\u03c9] maximal, \u21d4 in Fp has no solution to x^2+x+1?<\/p>\n\n\n\n<p>Update 12\/5: My assistants had to travel many moons and across many mountains to find a wise person who could answer this question. Here is his answer: The notation to me asks, &#8220;We are looking for solutions omega, in the complex plane that satisfy the polynomial P and want to know if they are, or are not deMoivre numbers, that is, complex numbers that when raised to an integer power (in this case 3 from the cube root) produce the value of 1. Both roots\u00a0of this polynomial satisfy this as I have shown. I&#8217;m not sure by Fp, whether they are referring to the function space of polynomials, the antiderivative, the derivative or a field relationship. I&#8217;m also not sure what they are looking for with respect to the prime constraint, all the coefficients are 1, 1 is defined NOT to be prime so I&#8217;m not sure how to satisfy that issue. For more info, try this online textbook on Algebraic Number Theory from Stanford U: <a href=\"http:\/\/bit.ly\/stanford-number-theory\">bit.ly\/stanford-number-theory<\/a><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" width=\"960\" height=\"704\" src=\"https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/learning-calculus.jpg\" alt=\"Proposed solution for the calculus problem, showing a graph of the equation y=1+x+x^2\" class=\"wp-image-11856\" srcset=\"https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/learning-calculus.jpg 960w, https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/learning-calculus-300x220.jpg 300w, https:\/\/library.bc.edu\/answerwall\/wp-content\/uploads\/2019\/12\/learning-calculus-768x563.jpg 768w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><figcaption>Proposed solution for the calculus problem.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>My advanced math assistants have not yet responded; I hope to have an answer tomorrow. In the meantime, I&#8217;d like to clarify whether I&#8217;ve represented your post accurately in typeset: \u03c9 \u2208 C cube root unity, P prime. Then (P) \u2208 Z[\u03c9] maximal, \u21d4 in Fp has no solution to x^2+x+1? Update 12\/5: My assistants &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/library.bc.edu\/answerwall\/2019\/12\/03\/cube-root-unity-in-x2x1\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;cube root unity in x^2+x+1&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[155,930],"class_list":["post-11777","post","type-post","status-publish","format-standard","hentry","category-academics","tag-math","tag-number-theory"],"_links":{"self":[{"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/posts\/11777","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/comments?post=11777"}],"version-history":[{"count":7,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/posts\/11777\/revisions"}],"predecessor-version":[{"id":11859,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/posts\/11777\/revisions\/11859"}],"wp:attachment":[{"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/media?parent=11777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/categories?post=11777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/library.bc.edu\/answerwall\/wp-json\/wp\/v2\/tags?post=11777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}