Cogalois TheoryThis volume offers a systematic, comprehensive investigation of field extensions, finite or not, that possess a Cogalois correspondence. The subject is somewhat dual to the very classical Galois Theory dealing with field extensions possessing a Galois correspondence. Solidly backed by over 250 exercises and an extensive bibliography, this book presents a compact and complete review of basic field theory, considers the Vahlen-Capelli Criterion, investigates the radical, Kneser, strongly Kneser, Cogalois, and G-Cogalois extensions, discusses field extensions that are simultaneously Galois and G-Cogalois, and presents nice applications to elementary field arithmetic. |
Contents
PREFACE | 1 |
PRELIMINARIES | 15 |
KNESER EXTENSIONS | 53 |
COGALOIS EXTENSIONS | 69 |
STRONGLY KNESER EXTENSIONS | 89 |
GALOIS GCOGALOIS EXTENSIONS | 125 |
RADICAL EXTENSIONS AND CROSSED | 153 |
EXAMPLES OF GCOGALOIS EXTENSIONS | 173 |
CONNECTIONS WITH GRADED ALGEBRAS | 229 |
INFINITE KNESER EXTENSIONS | 259 |
INFINITE GCOGALOIS EXTENSIONS | 269 |
INFINITE KUMMER THEORY | 283 |
INFINITE GALOIS THEORY | 291 |
INFINITE GALOIS GCOGALOIS | 305 |
| 329 | |
| 335 | |
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Common terms and phrases
Abelian extension Abelian group Albu and Nicolae algebraic number field arbitrary assertions are equivalent Assume Barrera-Mora canonical Chapter Char(F classical Kummer extension Cog(E/F Cogalois Corollary deduce denote E/F is G-Cogalois E/F is G-Kneser element exists extension E/F field extension field F finite extension finite Galois extension finite group following assertions following statements hold G-Kneser extension G-radical extension E/F G₁ Gal(E/F Galois extension Galois group Galois Theory gcd(n group G group G/F group isomorphism hence infinite integers Intermediate E/F intermediate field irreducible isomorphisms of lattices K*G-Kneser Kne E/F Kneser Criterion Kneser group Kummer extensions lattice isomorphism Lemma Let E/F Let F morphism n-Kummer notation odd prime polynomial prime number profinite group PROOF Proposition Prove Qr,d radical resp ring roots of unity Section separable extension set of representatives shows strongly G-Kneser subextension subfield subset Theorem vector space basis