Algebra
Volume I: Fields and Galois Theory
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ISBN-13:
9780387289304
Veröffentl:
2005
Erscheinungsdatum:
08.12.2005
Seiten:
296
Autor:
Falko Lorenz
Gewicht:
426 g
Format:
234x165x15 mm
Sprache:
Englisch
Beschreibung:
The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry.The main focus of the second volume is on additional structure of fields and related topics. Much material not usually covered in textbooks appears here, including real fields and quadratic forms, diophantine dimensions of a field, the calculus of Witt vectors, the Schur group of a field, and local class field theory.
From Math Reviews:This is a charming textbook, introducing the reader to the classicalparts of algebra. The exposition is admirably clear and lucidlywritten with only minimal prerequisites from linear algebra...The book ends with an appendix containing exercises and notes on theprevious parts of the book. However, brief historical comments andsuggestions for further reading are also scattered through the text.
Constructibility with Ruler and Compass.- Algebraic Extensions.- Simple Extensions.- Fundamentals of Divisibility.- Prime Factorization in Polynomial Rings. Gauss's Theorem.- Polynomial Splitting Fields.- Separable Extensions.- Galois Extensions.- Finite Fields, Cyclic Groups and Roots of Unity.- Group Actions.- Applications of Galois Theory to Cyclotomic Fields.- Further Steps into Galois Theory.- Norm and Trace.- Binomial Equations.- Solvability of Equations.- Integral Ring Extensions with Applications to Galois Theory.- The Transcendence of ?.- Fundamentals of Transcendental Field Extensions.- Hilbert's Nullstellensatz.
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