Antoine Chambert-Loir – författare

This book stems from lectures on commutative algebra for 4th-year university students at two French universities (Paris and Rennes). At that level, students have already followed a basic course in linear algebra and are essentially fluent with the language of vector spaces over fields. The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry.

The book also includes 11 historical vignettes about mathematicians who contributed to commutative algebra. 

This book provides an introduction to information theory, focussing on Shannon’s three foundational theorems of 1948–1949. Shannon’s first two theorems, based on the notion of entropy in probability theory, specify the extent to which a message can be compressed for fast transmission and how to erase errors associated with poor transmission. The third theorem, using Fourier theory, ensures that a signal can be reconstructed from a sufficiently fine sampling of it. These three theorems constitute the roadmap of the book. 

The first chapter studies the entropy of a discrete random variable and related notions. The second chapter, on compression and error correcting, introduces the concept of coding, proves the existence of optimal codes and good codes (Shannon''s first theorem), and shows how information can be transmitted in the presence of noise (Shannon''s second theorem). The third chapter proves the sampling theorem (Shannon''s third theorem) and looks at its connections with other results, such as the Poisson summation formula. Finally, there is a discussion of the uncertainty principle in information theory.

Featuring a good supply of exercises (with solutions), and an introductory chapter covering the prerequisites, this text stems out lectures given to mathematics/computer science students at the beginning graduate level.