{"id":"galois-theory","domain":"algebra","type":"Concept","title":"Galois Theory","slug":"galois-theory","url":"/mathematics/algebra/galois-theory/","summary":"Connects field extensions to group theory via the Galois group. A polynomial is solvable by radicals iff its Galois group is solvable. Explains why no general formula exists for degree-5 polynomials.","introduced_by":"Évariste Galois, 1830-1832; published posthumously by Liouville, 1846","key_results":[{"result":"Fundamental Theorem of Galois Theory","statement":"Subgroups of Gal(K/F) correspond bijectively, order-reversingly, to intermediate fields."},{"result":"Abel-Ruffini Theorem","statement":"No general radical formula exists for polynomial equations of degree 5 or higher."}],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Stewart, I. (2015). Galois Theory. 4th ed. CRC Press."},{"tier":1,"citation":"Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons.","chapter":"14"},{"tier":2,"citation":"Tignol, J-P. (2001). Galois' Theory of Algebraic Equations. World Scientific."},{"tier":3,"citation":"Livio, M. (2005). The Equation That Couldn't Be Solved. Simon & Schuster."}],"relationships":[{"type":"built_from","target":"group","label":"Group"},{"type":"built_from","target":"field","label":"Field"},{"type":"key_person","target":"galois-evariste","label":"Évariste Galois"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.3"}