By Antoine Chambert-Loir

This designated textbook specializes in the constitution of fields and is meant for a moment direction in summary algebra. along with delivering proofs of the transcendence of pi and e, the publication comprises fabric on differential Galois teams and an evidence of Hilbert's irreducibility theorem. The reader will listen approximately equations, either polynomial and differential, and concerning the algebraic constitution in their options. In explaining those ideas, the writer additionally offers reviews on their historic improvement and leads the reader alongside many attention-grabbing paths. additionally, there are theorems from research: as acknowledged earlier than, the transcendence of the numbers pi and e, the truth that the complicated numbers shape an algebraically closed box, and likewise Puiseux's theorem that exhibits how you can parametrize the roots of polynomial equations, the coefficients of that are allowed to change. There are routines on the finish of every bankruptcy, various in measure from effortless to tricky. To make the ebook extra energetic, the writer has included photographs from the historical past of arithmetic, together with scans of mathematical stamps and photographs of mathematicians. Antoine Chambert-Loir taught this e-book while he used to be Professor at École Polytechnique, Palaiseau, France. he's now Professor at Université de Rennes 1.

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