{"id":133,"date":"2019-02-02T05:20:40","date_gmt":"2019-02-02T05:20:40","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=133"},"modified":"2019-02-08T02:48:51","modified_gmt":"2019-02-08T02:48:51","slug":"the-binary-tetrahedral-group-as-a-p-adic-galois-group","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2019\/02\/02\/the-binary-tetrahedral-group-as-a-p-adic-galois-group\/","title":{"rendered":"The binary tetrahedral group as a p-adic Galois group."},"content":{"rendered":"

Let \"G\" be the binary tetrahedral group. This group appears as the double cover of the group of rotations of the tetrahedron (under \"SU(2)\to SO(3)\"), as a group of units in an appropriate \"\mathbb{Z}\"-form of the quaternion group, or as \"SL_2(\mathbb{F}_3)\".<\/p>\n

Consider a local field of residue characteristic \"p\". Now consider the Galois group of a finite Galois extension. It has a large pro-\"p\" part, together with two cyclic parts corresponding to the tamely ramified part and the unramified part. This structure alone shows that \"G\" cannot be such a group unless \"p=2\".<\/p>\n

Alternatively, every 2-dimensional irreducible representation of the Galois group of a local field of odd residue characteristic is induced, and \"G\" has no index two subgroups, so again cannot occur as a Galois group when \"p\" is odd.<\/p>\n

So what about when \"p\" is even? By considering the fixed field of a Sylow-2-subgroup, we see that if \"G\" appears as a Galois group, then it is the Galois closure of a degree 8 extension.<\/p>\n

Now the degree 8 extensions of \"\mathbb{Q}_2\" are finite and number and have all been computed, together with their Galois groups. They can be found at this online database<\/a>. A quick examination shows that our binary tetrahedral group does not appear.<\/p>\n

But it does appear as an inertia group. So \"G\" is not a Galois group over \"\mathbb{Q}_2\", but is a Galois group over the unramified quadratic extension of \"\mathbb{Q}_2\".<\/p>\n","protected":false},"excerpt":{"rendered":"

Let be the binary tetrahedral group. This group appears as the double cover of the group of rotations of the tetrahedron (under ), as a group of units in an appropriate -form of the quaternion group, or as . Consider … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-133","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":false,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-29","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/133","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=133"}],"version-history":[{"count":4,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/133\/revisions"}],"predecessor-version":[{"id":140,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/133\/revisions\/140"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}