Algebraic Extensions of FieldsCourier Corporation, 7 janv. 2014 - 192 pages "...clear, unsophisticated and direct..." — Math This textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamental theorum of Galois theory and some examples, it contains discussions of cyclic extensions, Abelian extensions (Kummer theory), and the solutions of polynomial equations by radicals. Chapter 2 concludes with three sections devoted to the study of infinite algebraic extensions. The study of valuation theory, including a thorough discussion of prolongations of valuations, begins with Chapter 3. Chapter 4 is concerned with extensions of valuated field, and in particular, with extensions of complete valuated fields. Chapter 5 contains a proof of the unique factorization theorum for ideals of the ring of integers of an algebraic number field. The treatment is valuation-theoretic throughout. The chapter also contains a discussion of extensions of such fields. A special feature of this book is its more than 200 exercises - many of which contain new ideas and powerful applications - enabling students to see theoretical results studied in the text amplified by integration with these concrete exercises. |
Table des matières
| 1 | |
The fundamental theorem of Galois theory | 36 |
The first cohomology group | 43 |
Additive Kummer theory | 52 |
Infinite Galois extensions | 58 |
Introduction to Valuation Theory | 73 |
Extensions of Valuated Fields | 105 |
Dedekind Fields | 132 |
Proof of Theorem 19 of Chapter 2 | 159 |
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Expressions et termes fréquents
Abelian algebraic arbitrary assume called Cauchy sequence Chapter char closed coefficients consider contains COROLLARY cyclic cyclic extension define denote determine distinct divides edition equations example Exercise extension of k fact factors field and let finite extension finite number first follows functions Galois extension given Hence holds homomorphism implies inseparable introduction Irr k,a isomorphism k-isomorphism lemma Let f Let K linearly mapping mathematics non-Archimedean valuated nonzero prime ideal normal extension obtain polynomial polynomial in k[x positive integer prime element problems prolongation Proof PROPOSITION prove quantum mechanics ramified rational respect result ring of integers root of unity separable separable extension Show solutions solvable splitting field subfield subgroup Suppose Theorem theory topology unique unit unramified valuated field write zero