Algebraic Extensions of Fields (eBook)

Paul J. McCarthy (Autor)

eBook Download: EPUB

2014
192 Seiten
Dover Publications (Verlag)
978-0-486-78147-1 (ISBN)

Infos

Graduate-level coverage of Galois theory, especially development of infinite Galois theory; theory of valuations, prolongation of rank-one valuations, more. Over 200 exercises. Bibliography. "...clear, unsophisticated and direct..." — Math.

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"e;...clear, unsophisticated and direct..."e; — MathThis 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.

Produkt-Details

• Erscheint lt. Verlag: 7.1.2014
• Sprache: englisch
• Maße: 140 x 140 mm
• Gewicht: 209 g
• Themenwelt: Mathematik / Informatik ► Mathematik
• Schlagworte: Algebra • Algebraic fields • Mathematics
• ISBN-10: 0-486-78147-X / 048678147X
• ISBN-13: 978-0-486-78147-1 / 9780486781471