3-class field towers with 2 or 3 stages

3-class field towers with 2 or 3 stages | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article 3-class field towers with 2 or 3 stages Helga Boyer von Berghof, Daniel C. Mayer This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8408859/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract For quadratic fields k=Q(d^1/2) with discriminant d, 3-class group Cl(3,k) =(Z/3Z)^2, and one of six principalization types kappa(k) in {(1122),(2122),(3122),(1231),(2231),(4231)}, we seek necessary and sufficient conditions for the Galois group S=Gal(F(3,infinity,k)/k) of the unramified Hilbert 3-class field tower of k to coincide with the Galois group M=Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k.In the case of non-coincidence,we study the path between M and S in the descendant tree of the elementary bicyclic 3-group (Z/3Z)^2. Minimal discriminants d with assigned principalization type kappa(k) and fixed length ell(3,k) in {2,3} of the 3-class field tower are determined experimentally for nilpotency class 5 <= cl(M) <= 8. 2000 Mathematics Subject Classification. 11R37, 11R29, 11R11, 11R16, 11R20; 20D15, 20F14. 3-class field tower metabelianization second 3-class group unramified cyclic cubic extensions principalization of 3-classes abelian type invariants quadratic fields cubic fields dihedral fields Artin reciprocity law group transfer kernels abelian quotient invariants of first and second order relation rank Shafarevich theorem balanced presentation inversion automorphism Schur σ-groups central series descendant trees. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8408859","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":564553339,"identity":"4902a803-83af-48c7-9f96-16d2569106af","order_by":0,"name":"Helga Boyer von Berghof","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Helga","middleName":"Boyer","lastName":"von Berghof","suffix":""},{"id":564553341,"identity":"44bbb0c3-839d-49c4-91f0-4672ed4397c6","order_by":1,"name":"Daniel C. 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Minimal discriminants d with assigned principalization type kappa(k) and fixed length ell(3,k) in {2,3} of the 3-class field tower are determined experimentally for nilpotency class 5 \u0026lt;= cl(M) \u0026lt;= 8.\u003c/p\u003e\n\u003cp\u003e2000 Mathematics Subject Classification. 11R37, 11R29, 11R11, 11R16, 11R20; 20D15, 20F14.\u003c/p\u003e","manuscriptTitle":"3-class field towers with 2 or 3 stages","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-24 05:21:41","doi":"10.21203/rs.3.rs-8408859/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e418d16b-d04a-44f8-9f31-6c5c57057d27","owner":[],"postedDate":"December 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-02T10:39:37+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-24 05:21:41","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8408859","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8408859","identity":"rs-8408859","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}