{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T17:02:07Z","timestamp":1781456527187,"version":"3.54.1"},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we establish the Galois connection between the intermediate fields of an extension\n                    <jats:italic>E<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    and the subgroups of the automorphism group Aut(\n                    <jats:italic>E, F<\/jats:italic>\n                    ). We show that if\n                    <jats:italic>E<\/jats:italic>\n                    is a finite Galois extension of\n                    <jats:italic>F<\/jats:italic>\n                    , then this connection induces a bijection between all intermediate fields of\n                    <jats:italic>E<\/jats:italic>\n                    and\n                    <jats:italic>F<\/jats:italic>\n                    and all subgroups of Aut(\n                    <jats:italic>E, F<\/jats:italic>\n                    ).\n                  <\/jats:p>","DOI":"10.2478\/forma-2026-0001","type":"journal-article","created":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T16:46:59Z","timestamp":1781455619000},"page":"1-8","source":"Crossref","is-referenced-by-count":0,"title":["The Galois Connection between IntermediateFields(E,F) and Subgroups of Aut(E,F)"],"prefix":"10.2478","volume":"34","author":[{"given":"Christoph","family":"Schwarzweller","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Gda\u0144sk Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2026,6,13]]},"reference":[{"key":"2026061319002683202_j_forma-2026-0001_ref_001","unstructured":"Emil Artin. Galois Theory: Lectures Delivered at the University of Notre Dame, volume 2 of Notre Dame Mathematical Lectures. Dover Publications, Mineola, NY, 1998. ISBN 9780486623429."},{"key":"2026061319002683202_j_forma-2026-0001_ref_002","doi-asserted-by":"crossref","unstructured":"Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8 17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026061319002683202_j_forma-2026-0001_ref_003","doi-asserted-by":"crossref","unstructured":"Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026061319002683202_j_forma-2026-0001_ref_004","doi-asserted-by":"crossref","unstructured":"Thomas Browning and Patrick Lutz. Formalizing Galois theory. Experimental Mathematics, 31(2):413\u2013424, 2022. doi:10.1080\/10586458.2021.1986176.","DOI":"10.1080\/10586458.2021.1986176"},{"key":"2026061319002683202_j_forma-2026-0001_ref_005","unstructured":"David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley and Sons, third edition, 2004."},{"key":"2026061319002683202_j_forma-2026-0001_ref_006","doi-asserted-by":"crossref","unstructured":"Rod Gow and Rachel Quinlan. Galois theory and linear algebra. Linear Algebra and Its Applications, 430(7):1778\u20131789, 2009. doi:10.1016\/j.laa.2008.06.030.","DOI":"10.1016\/j.laa.2008.06.030"},{"key":"2026061319002683202_j_forma-2026-0001_ref_007","doi-asserted-by":"crossref","unstructured":"Adam Grabowski, Artur Korni\u0142owicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363\u2013371, 2016. doi:10.15439\/2016F520.","DOI":"10.15439\/2016F520"},{"key":"2026061319002683202_j_forma-2026-0001_ref_008","unstructured":"I. Martin Isaacs. Algebra: A Graduate Course. Wadsworth Inc., 1994."},{"key":"2026061319002683202_j_forma-2026-0001_ref_009","unstructured":"Serge Lang. Algebra (Revised Third Edition). Springer Verlag, 2002."},{"key":"2026061319002683202_j_forma-2026-0001_ref_010","doi-asserted-by":"crossref","unstructured":"Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language. In Automated Deduction \u2013 CADE 28: 28th International Conference on Automated Deduction, Virtual Event, July 12\u201315, 2021, Proceedings, pages 625\u2013635, Berlin, Heidelberg, 2021. Springer-Verlag. doi:10.1007\/978-3-030-79876-5 37.","DOI":"10.1007\/978-3-030-79876-5_37"},{"key":"2026061319002683202_j_forma-2026-0001_ref_011","unstructured":"Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."},{"key":"2026061319002683202_j_forma-2026-0001_ref_012","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller. Field extensions and Kronecker\u2019s construction. Formalized Mathematics, 27(3):229\u2013235, 2019. doi:10.2478\/forma-2019-0022.","DOI":"10.2478\/forma-2019-0022"},{"key":"2026061319002683202_j_forma-2026-0001_ref_013","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller and Agnieszka Rowi\u0144ska-Schwarzweller. Simple extensions. Formalized Mathematics, 31(1):287\u2013298, 2023. doi:10.2478\/forma-2023-0023.","DOI":"10.2478\/forma-2023-0023"},{"key":"2026061319002683202_j_forma-2026-0001_ref_014","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller and Agnieszka Rowi\u0144ska-Schwarzweller. Algebraic extensions. Formalized Mathematics, 29(1):39\u201348, 2021. doi:10.2478\/forma-2021-0004.","DOI":"10.2478\/forma-2021-0004"},{"key":"2026061319002683202_j_forma-2026-0001_ref_015","doi-asserted-by":"crossref","unstructured":"Christoph Schwarzweller and Agnieszka Rowi\u0144ska-Schwarzweller. Introduction to Galois theory. Formalized Mathematics, 33(1):175\u2013183, 2025. doi:10.2478\/forma-2025-0014.","DOI":"10.2478\/forma-2025-0014"},{"key":"2026061319002683202_j_forma-2026-0001_ref_016","doi-asserted-by":"crossref","unstructured":"Ian Stewart. Galois Theory. Chapman and Hall\/CRC, fourth edition, 2015.","DOI":"10.1201\/b18187"},{"key":"2026061319002683202_j_forma-2026-0001_ref_017","doi-asserted-by":"crossref","unstructured":"Steven H. Weintraub. Galois Theory. Springer-Verlag, second edition, 2009.","DOI":"10.1007\/978-0-387-87575-0"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2026-0001","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T16:48:56Z","timestamp":1781455736000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2026-0001"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,1]]},"references-count":17,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,6,13]]},"published-print":{"date-parts":[[2026,6,1]]}},"alternative-id":["10.2478\/forma-2026-0001"],"URL":"https:\/\/doi.org\/10.2478\/forma-2026-0001","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,1]]}}}