A Field Guide to Algebra [electronic resource] / by Antoine Chambert-Loir.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Undergraduate Texts in Mathematics: Publisher: New York, NY : Springer New York, 2005Description: X, 198 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780387269559Subject(s): Mathematics | Algebra | Commutative algebra | Commutative rings | Field theory (Physics) | Number theory | Mathematics | Algebra | Field Theory and Polynomials | Number Theory | Commutative Rings and AlgebrasAdditional physical formats: Printed edition:: No titleDDC classification: 512 LOC classification: QA150-272Online resources: Click here to access online
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Field extensions -- Roots -- Galois theory -- A bit of group theory -- Applications -- Algebraic theory of differential equations.
This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. The reader will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads the reader along many interesting paths. In addition, there are theorems from analysis: as stated before, the transcendence of the numbers pi and e, the fact that the complex numbers form an algebraically closed field, and also Puiseux's theorem that shows how one can parametrize the roots of polynomial equations, the coefficients of which are allowed to vary. There are exercises at the end of each chapter, varying in degree from easy to difficult. To make the book more lively, the author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians. Antoine Chambert-Loir taught this book when he was Professor at École polytechnique, Palaiseau, France. He is now Professor at Université de Rennes 1.