{"id":40,"date":"2018-01-08T16:35:41","date_gmt":"2018-01-08T16:35:41","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=40"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"p-extensions","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/p-extensions\/","title":{"rendered":"Extensions of the $$p$$-adic field $$\\mathbb{Q}_p$$ with Galois group $$E_1$$"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-41 size-full\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/e1-subgroup.gif\" alt=\"Extensions of the $$p$$-adic field $$\\mathbb{Q}_p$$ with Galois group $$E_1$$\" width=\"748\" height=\"378\" \/><\/p>\n<p>Let $$p$$ be an odd prime number. The graph shows the subgroup lattice of the group $$E_1$$, which is the unique non-abelian group of order $$p^3$$ of exponent $$p$$. $$C_p$$ denotes the cyclic group with $$p$$ elements.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-42 size-full\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/e1-subfield.gif\" alt=\"Extensions of the $$p$$-adic field $$\\mathbb{Q}_p$$ with Galois group $$E_1$$\" width=\"837\" height=\"427\" \/><\/p>\n<p>The subfield lattice of the unique extension of $$\\mathbb{Q}_p$$ with Galois group $$E_1$$ with the minimal polynomials for the generating elements of the ramified (sub)extension of degree $$p$$. Furthermore inertia degrees $$(f=p)$$ and ramification indices $$(e=p,~E=p)$$ are given. A proof can be found in the thesis <a href=\"https:\/\/math-sites.uncg.edu\/sites\/pauli\/publications\/pauli_phd-thesis.pdf\">Efficient Enumeration of Extensions of Local Fields with Bounded Discriminant<\/a> by <a href=\"http:\/\/www.uncg.edu\/mat\/people\/people.php?username=s_pauli\">Sebastian Pauli<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let $$p$$ be an odd prime number. The graph shows the subgroup lattice of the group $$E_1$$, which is the unique non-abelian&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-40","post","type-post","status-publish","format-standard","hentry","category-number-theory"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/40","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/comments?post=40"}],"version-history":[{"count":4,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/40\/revisions"}],"predecessor-version":[{"id":243,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/40\/revisions\/243"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=40"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=40"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}